The earliest match for this sense of the word that I find in an Elephind search is from Edwin Goodgold, "Joe Nalven: A Champion's Psyche," in the Columbia [University, New York City] Daily Spectator (March 25, 1965):
His [Nalven's] strongest competition came from a short and pesky foilsman who went to Penn [the University of Pennsylvania]. This little guy tried to drive his opponents mad with an unorthodox style which featured his jumping up and down like a baby kangaroo, and bobbing and weaving like a boxer. The imp's opponents usually lost their patience, and, getting increasingly ticked off at his irregular behavior, they usually lunged at him hoping to make short work of this nemesis. Unfortunately for them, the Penn fencer was quite an alert parrier, and they usually discovered that the point of his blade was in their navel before their weapons had got anywhere near him.
Robert Chapman & Barbara Kipfer, Dictionary of American Slang third edition (1995) has this entry for the phrase:
ticked off or ticked adj phr or adj by 1959 Angry; =PISSED OFF, TEE'D OFF: Steve Kemp is ticked off—Village Voice
Interestingly, the same source claims that related expression "tick [someone] off" goes back to the early 20th century:
tick [someone] off v phr by 1915 To anger someone; PISS someone OFF. Just the slightest thing could tick off Harold—Philadelphia
However, the earliest predecessor of this dictionary to notice either expression is Harold Wentworth & Stuart Flexner, Dictionary of American Slang, supplemented edition (1967), which offers these entries in the supplement:
ticked off = tee'd off.
tick [someone] off To annoy; to make angry. Usu. in the passive.
This dictionary reproduces the entry for "tee'd off" that originally appeared in Wentworth & Flexner, Dictionary of American Slang, first edition (1960):
tee'd off[;] teed off[;] t'd off adj. Angry, fed up; disgusted. See pissed off. From "tee-off" reinforced by "pee'd off."
'Ticked him off'
An Elephind search for "ticked him off" turns up some interesting but ambiguous results from thwe 1910s. Notably, from "Girls Who Talk Too Much," in the Adelaide [South Australia] Chronicle (October 8, 1921):
There is an irritating quality in Miss Echo's recitals, for, when she begins to grow at all interesting and the whole carriageful of people are at last irritated into wanting to hear what she really did say to Arthur when she 'ticked him off,' her voice is lost in a giggle.
But this item is from the wrong continent—and it isn't clear that the expression a used here doesn't have a very different meaning than "angered him"; my guess is that it may mean something closer to "wrote him off" or "rejected him" or perhaps "presented him with an itemized account of his demerits." Indeed, a kindred usage appears two years earlier in "Isles Without Law," in the [Sydney, New South Wales] Farmer and Settler (June 3, 1919):
Eve had done much island travel by this time, and knew just what to expect of a voyage. So the feelings of James Evans, the Island Manager, were not such a sealed book to the Coral Queen as Mr. Evans might have supposed. In fact, Eve had already ticked him off as "flirty, but won't get mushy; safe to use." And she proposed to use him.
Here, it seems fairly clear that the sense of the expression is "appraised him and determined his status." In my view, this Australian usage and the considerably later U.S. usage are unlikely to be related. I didn't find any instances in Elephind search results of "ticked him off" from the period 1900–1940 that had clearly had the sense "angered him."
By the 1930s the meaning of the phrase encompassed the sense "criticized him" or "took him to task," and we have this later example from Agatha Christie, "A Pocketful of Rye," serialized in the [Sydney, New South Wales] Sun-Herald (November 15, 1953) where it may mean something like "set him off":
"Still, there would have to be a motive, I suppose."
"I really don't know that that sort of person requires much motive. I dare say Mr. Fortescue ticked him off about something, and I rather suspect that sometimes he drinks too much. But what I really think is that he's a bit unbalanced, you know. Like that footman, or butler, whoever it was, who went round the house shooting everybody. Of course, to be quite honest with you, I did suspect that it was Adele who poisoned Mr. Fortescue. But now, of course, one can't suspect that since she's been poisoned herself. ..."
And in this example from "Gina Lollobrigida Tells Her Own Story," in the [Perth, Western Australia] Mirror (December 17 1955), it clearly has the meaning "criticized him," leading immediately (in this case to "He lost his temper":
While we were dancing, Bob started grumbling about my making him take the floor. I ticked him off. He lost his temper, and in the next minute we were both shouting at each other.
Ultimately I couldn't find any evidence from the United States that "ticked [someone] off" goes back to the 1910s. The evolution of Australian and perhaps British usage is quite interesting but doesn't provide a basis for concluding that "tick [someone] off" and "ticked off" in the sense of "angered [someone]" and "angry" are substantially older in U.S. usage than the late 1950s.