The earliest matches for "winning streak" that appear in Elephind newspaper database searches come from the 1880s and involve the sport of baseball. Of the 26 unique matches that Elephind returns for "winning streak" from before 1890, 25 involve baseball and 1 horse racing.
Paul Dickson, The Dickson Baseball Dictionary (1989) has this rather odd entry for the expression:
winning streak n. Consecutive victories by a team as an individual pitcher.
1ST [occurrence] 1897. (New York Tribune, August 3; EJN)
"EJN" is Edward J. Nugent, Historical Dictionary of Baseball Terminology (1939).
Two victories in a row certainly meets the technical definition of "consecutive victories," so it would seem that, back-to-back wins in baseball should suffice to constitute a (very modest) winning streak. By the same standard, however, one victory should not. In retrospect, if the team or starting pitcher won the next game played, you might reasonably argue that the first win marked the start of what later proved to be a winning streak.
The earliest match for "winning streak" from my Elephind search involves just such a two-game streak. From an untitled brief item in the Champaign [Illinois] Daily News (August 24, 1885):
—The Deers ball nine have struck a winning streak at last. Saturday they defeated Sidney by a score of 11 to [4(?)], and Tuesday they beat Blackberry 35 to [6(?)].
(Unfortunately, although Elephind includes the entire excerpt in a search results snippet, the newspaper itself is inaccessible for viewing.)
But another early instance suggests that the notion of "entering upon a winning streak" may have been viewed as something of an aspirational state of mind. From "Base Ball News" in the Lancaster [Pennsylvania] Daily Intelligencer (August 12, 1886):
Speaking of the Athletic club, their scorer says through the Press "Miller's victory yesterday has inspired the team and himself to enter upon a winning streak." It is nearly time for something of the kind to happen, as the club is thirteen games behind the next highest.
There is no clear indication here that Miller and the Athletics had won more than one consecutive game at the time of the report.
A number of early accounts of baseball games use the expression "struck a winning streak"—in some instances where we might today expect to see "started a winning streak." One especially interesting instance involves game 7 of the 10-game World Series of 1888 between the St. Louis Browns and the New York Giants. After splitting the first two games of the series, New York won games 3, 4, 5, and 6. But after St. Louis broke its losing streak at four games, the St. Paul [Minnesota] Daily Globe (October 25, 1888) offered this headline:
Turning the Tables: Von Der Ahe's Browns Strike a Winning Streak at St. Louis
That can only refer to a winning streak of one game, since game 8 (which St. Louis lost) was played on the day of the 25th.
On the other hand, less than 60 days earlier, the term "winning streak" had appeared in an article in the New York Sun (September 2, 1888) in the context of a skein of victories that even a person with a very high threshold requirement would acknowledge qualifies:
The Buffalos suddenly struck a winning streak and won thirteen straight. Among this number were two victories at Syracuse.
Whatever the earliest understanding of "winning streak" may have been, it seems clear that today either the reality or the expectation of two or more consecutive wins is a prerequisite for claiming a streak or the start of a streak.