The origin of the word “bingo” seems to pose a classic chicken-or-egg question: is the game called “Bingo” because game winners shout it, or do folks shout “bingo!” as an interjection in other situations (expressing, as the American Heritage Dictionary puts it, “the sudden completion of an event, occurrence of an idea, or confirmation of a guess”) in imitation of winning the game Bingo?
Unfortunately, there is no clear winner here. The two uses (game name and interjection) appeared in print at roughly the same time, the interjection “bingo” in 1927 and the game name in 1936. An argument in favor of the game coming first is that Bingo itself is a form of Lotto, which dates back to at least the 18th century. On the other hand, “bing” has a history as indicating sudden action since the 1920s (”Now I do this kind of thing On the wing, on the wing! Bing!”, James Joyce, Ulysses). This “bing” is almost certainly “echoic” in origin, meant to imitate the sound of sudden impact or explosion. “Bingo” is also a 17th century slang term for brandy (as in “stingo and bingo,” strong ale and brandy).
My suspicion is that the interjection “bingo” came first, growing out of “bing,” and was adopted as the name of the game because (a) winning is sudden and exciting, and (b) the game resembles Lotto, making the “o” ending appropriate. But the game of Bingo certainly popularized “bingo” as an interjection, so, at least in that sense, everybody wins.