The exact phrase "put one over on" in the sense of "get the better of"—through superior skill, superior strategy (or trickery), or the element of surprise—appears to have caught on quickly in the United States, emerging in the early 1900s and becoming very popular by 1910.
The earliest matches that I've been able to find come from the period 1903 to 1905 from various sports milieus—specifically, boxing, baseball, motorboat racing, and motorcar racing. From "Doings in the Square Circle," in the [Salt Lake City, Utah] Deseret Evening News (May 9, 1903):
A report from New York has it that Rob Fitzsimmons will train [Jim] Jeffries for the mill with [Jim] Corbett. It would add interest to the fight should Fitzsimmons take a hand one way or the other, and it would not be surprising to the public to see the Cornishman in Jeffries' corner telling the big fellow how to put one over on the cleverest of the heavyweights. In fact, it teems like a reasonable arrangement.
From "Jake Lost Chance to Buy Nationals," in the Washington [D.C.] Times (May 23, 1905):
Recently in talking to [National League] President [Ban] Johnson relative to the success of [Jake] Stahl as manager of the Nationals [Washington's baseball team], [Bob] McRoy said "We came pretty near putting one over on you down there that you never heard of."
"What was that," queried the interested president.
"Stahl and I got the funds and thought of buying the stock held by the league if that option had expired."
"Well, well," said Johnson, "I never thought of that combination, but if you had spoken of it to me, I could have put you in right, even after that option was taken up. Several of the minority stockholders wanted to sell out, too, and you could have bought their shares for about $2 apiece. Now the tam is doing so good you couldn't get them for $20."
From "A Warm Game: Barbers and Railway Employees Beat Clerks" in the [Greenfield, Indiana] Evening Star (July 26, 1905):
A combination ball team picked from the barbers and the street railway employees put one over on the clerks yesterday. At least, that seems to be the case.
In the heat of the excitement which immediately followed the game, the official score keeper fainted away, but upon his person was found a small, folded piece of paper, bearing the following hasty inscription, "Barbers ten, Clerks four, / I score no———!"
From "Europe Finishes Year's Calendar," in [Chicago, Illinois] Motor Age (October 5, 1905):
France can well be satisfied with her record. The Bennett cup race, won by Thery, redounds to the credit of her makers, especially the Richard-Brasier people, while in the classic Ardennes circuit race French colors were again home first in Hemery's Darracq car. The great Pyrennes tour, vying as it did with Germany's Herkomer, was an unqualified success, M. Sorel vindicating ht merits of the De Dietrich, sixty-four racers taking part in the long trip. While France was not officially represented in the Harmsworth cup contest for the motor boat championship, she put one over on the English in the cross channel race in which La Rapiere succeeded in gaining the verdict over the English Napier.
From "Jump Sparks," in [Chicago, Illinois] Motor Age (October 12, 1905):
In forcing the French makers to recognize the right of the English manufacturer to bid first for the business to be found in the United Kingdom by compelling them to exhibit at the Olympia show, John Bull has sort of put one over on his friend, the enemy across the channel.
Instances of the expression began to proliferate in 1906, again mostly in sporting contexts but occasionally elsewhere as well.
From "Clunie Wants Joe Corbett: Manager Hanlon of the Reds, However, Demands $3000 for Release of Pitcher," in the San Francisco [California] Call (February 18, 1906):
Clunie now wants Corbett. He had a talk with Joe a few days ago and began to frame up terms. Corbett wired Hanlon to find out about the proposition and discovered that the Cincinnati manager demanded $3000 for his release.
This is a big sum to pay for any ball player and consequently when Clunie received the news he was under the impression that Hanlon was trying to "put one over" on him. He is fuming and raging about this at the present time, but cannot see any way out of it right now.
From "Stop Knocking; Season's Young: Somebody Must Lose the Early Games," in the Washington [D.C.] Times (April 17, 1906):
Charley Jones put one over on the knockers by smashing a double to right in the sixth round, just when everybody was telling his neighbor that the Jones young man couldn't hit a river by jumping off a bridge. Jones also sacrificed once. His other efforts were weak, but cut little ice, as there was nobody on bases.
From Fred Mulholland, "Seattle Ring Remains Open: Bookmakers Find Money Plentiful at The Meadows," in the San Francisco [California] Call (June 30, 1906):
While many of the other owners were stargazing George McNeil had his horses ready for the opening and got away with a rush. Funnyside won three races right off the reel, Crigil two, Jackful one and Fury one.
Al Dobson, owner of Epicure, tried to put one over on the opening day with Creedmore. The plater's odds fell from 20 to 3, and he led to the stretch. There a spread foot retired him to the rear division.
From "Blow to Our Gretna Green: Jerry Ellis Though[t] Hammond Nice Place for Quiet Wedding" in the [Hammond, Indiana] Lake County Times (July 20, 1906):
With this kind of an outlay to deal with, Jerry and the late fingernail polisher [his bride] figured the dope that they could slip over to Crown Point, get the license, come to Hammond, and get married, glide out of the city in their touring car and none be the wiser. It was a pretty plan and it nearly broke our hearts to spoil it. Incidentally in inferentially disparaging of Hammond having a newspaper that gets the news and gets it quickly and accurately, and publishes it when it gets it, Jerome A. Ellis discounted the fact that he was so well known and tried to put one over on us. It was without doubt the gentleman's intention to spring the story at a dinner at the club some time hence, of how he outwitted the newspapers and the people and got married over among the Hoosiers. Too bad, Jerry, the bubble is punctured.
From "McCafferty Puts One Over on Bookies," in the San Francisco [California] Call (July 24, 1906):
NEW YORK. July 23.—John J. McCafferty, the horseman who was ruled off the turf last winter by the California Jockey Club for alleged "doping" of the Huguenot,played an important part in the result of the fifth race. He ws not at the track, but he engineered the coup on Flaxman, which cost the ring $60,000.
Flaxman was at odds of 20 to 1 at post bugle call and 10 o 1 when the horses reached the starting point, having been backed to that figure from 50 to 1. In a few books as high as 100 o 1 was offered against him. Flaxman won in a driving finish by a neck.
From "Stagg Has Bunch of Rule Tricks: First Line of Theoretical Football Outlined by Sage of Midway," in the Minneapolis [Minnesota] Journal (August 27, 1906):
Chicago, Aug. 27.—Coach A. A. Stagg of the maroons has "put one over" on his rivals in the coaching line, in the opinion of the football populace of the University of Chicago. When the maroon's director leaves shortly for a vacation he will take with him a good-sized bunch of diagrams and a till bigger lot of new plays for the fall campaign that he has worked out this summer in his football class, a unique institution, which in the parlance of its members, "has them all skinned."
From "News and Notes of Sport," in the Palestine [Texas] Daily Herald (September 17, 1906):
The Erie Pa club is another base ball organization with an alliterative line-up. With Crane, Clark, Cranston and Cole, Erie put one over on Pittsburg[h] recently to the tune of 4 to 3.
From "The Chicago Situation" in The [Indianapolis, Indiana] Clay-Worker (October 1906):
Speaking of big contracts, a young man of this city [Chicago] with a reputation for success, energy and shrewdness, who is on the road a good deal representing a large company in the trade, put one over on himself the other day, and the good story is being quietly circulated by his "loving friends." It seems he went to another city to close up a deal. He wired news of his success in this, and in writing confirmation added to his letter this cheering news:
""I have before me contract for 400,000 square feet, etc."
When this reached the general offices there was some gasping. The young salesman is known as "Whirlwind Johnny" and all that, but a contract of this size from the particular corner of the map from which he had written seemed almost impossible. On top of their astonishment, however, came a telegram announcing that the deal had been closed for "400,000 brick." Then the G. M. sat him down and dictated as sarcastic a letter as Colonel Jim Stuart ever let through the Chicago postoffice.
From "How She 'Put One Over' on the Family," in the Washington [D.C.] Times (October 21, 1906):
"Am not coming home tonight. Look for the reason in tomorrow morning's papers.—Jessie Morse Woods," is the card received late in the afternoon by Mr. and Mrs. F. W. Burr, who live in Monrovia, Cal., the uncle and aunt of the writer.
Mrs. Woods, then Miss Morse, left for Los Angeles ostensibly for the purpose of having her picture taken. It is hardly possible that the surprise she planned for her kinfolk will equal her own when she finds that her picture was undoubtedly taken and appears in the morning papers beside her wedding announcement.
From "Conscientious Rustic 'Puts One Over' on a Burglar," in the Washington [D.C.] Times (November 11, 1906):
John Wipf, a respected farmer living a few miles west of Omaha, is plaintiff in an action against John Smith, a "yeggman," who is said to be known to the police of nearly every important city in this country.
From "Freezes His Former Friend Out of an Interest in the Tuxedo Saloon: Blacksmith Backed by a Rich Liquor Firm," in the San Francisco [California] Call (November 23, 1906):
According to the story that is going the rounds of Fillmore street, [boxing promoter Eddie] Graney and [Harry] Corbett were supposed to have formed a partnership to run the Tuxedo saloon between them, with a prominent liquor firm to back them up. But, so the tale goes, Graney "put one over" on Corbett at the proverbial eleventh hour and succeeded in installing himself as master of ceremonies of the Tuxedo and leaving Corbett out in the cold.
Conclusions
It seems noteworthy that four of the earliest sixteen occurrences of "put one over on" in the Google Books and Elephind databases come from a particular San Francisco newspaper and that another four come from a particular Washington, D.C. newspaper. It also seems not at all coincidental that twelve of the earliest sixteen matches—including the eight oldest—for the expression in the relevant sense are from sports-related contexts.
Nevertheless, scarcely three years after the earliest instance I found (from May 1903), the expression appears in a non-sporting context—the semi-elopement of a Chicago celebrity, as reported in an Indiana newspaper (in July 1906).
The exact sense of "put one over on" varies across the different examples. Several (including the oldest) can be read as meaning simply "defeat," but others (including the second-oldest, from May 1905) convey a sense less of "defeat" than "surprise," "trick," or "pull a fast one on." The latter appears to be the primary sense of the expression today, to judge from this entry in Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) for the closely related phrase "put [X] over on":
put over on. Fool, deceive, as in We can't put anything over on Tom. {Early 1900s}
I note in clossing that the earliest instance of "put anything over on" in Elephind search results is from the Hammond [Indiana] Times (October 2, 1907) and the earliest instance of "put something over on" is from the Sacramento [California] Union (July 19, 1908). It thus appears that "put one over on" is slightly older than either of those two expressions in U.S. newspaper use. A Google Books search turns up a story by William Hard, "The Wisdom of the Night," in Handbook of the Chicago Industrial Exhibit (March 11–17, 1907) that includes the phrase "put anything over on":
"Say, kid," said I, "why did you do it?"
"Oh, that's all right," says he: "I stand up for everybody. None o' these coppers can put anything over on me. I'd a got that dope-fiend off, too, if he'd had any sense."
But that, too, is a bit later than the earliest instances I found of "put one over on."