I’ve always believed this was attributed to the great, late Casey Stengal, the long time manager of the New Yawk Yankees.
Would you be referring to this column?
According to The Phrase Finder,
This info is backed up by The Yale Book Of Quotations, which gives the date it was in the paper as March 10, 1976.
“It ain’t over 'til it’s over” is often attributed to Yogi Berra. I don’t find anything about a fat lady from either Berra or Stengel.
One of my favorite Casey Stengel quotes is “Good pitching will always stop good hitting and vice-versa.”
There are also claims that it is an offshoot of of an old Southern saying, “Church ain’t over 'til the fat lady sings”, but I haven’t run that one to ground yet.
Whether there’s any validity to my memory, I always thought it applied to the character of Brunhilde in one of the Wagner operas. Either that or Kate Smith. I also realized Yogi was shooting for the “fat lady” thing and did your basic Yogi misfire on it. The “Church ain’t over 'til the fat lady sings” would be (in my estimation) similar to the miscue from Yogi, since skinny ladies also sing in church.
Kate Smith’s sign-off tune was IIRC “When the moon comes over the mountain” or something like that, and Kate rarely pushed away from the table too soon.
The trouble is that there is no record of Yogi ever saying anything close to the phrase in question.
Be that as it may, you do find “it ain’t over till it’s over” among those things attributed to Yogi. For instance: Yogi Berra - Wikipedia
The book Southern Words And Sayings(January 1976), by Fabia Rue Smith and Charles Rayford Smith, has in it the saying “Church ain’t over till the fat lady sings”. Even if this was something they made up instead of a real Southern saying, it still predates the March 10th 1976 reference by at least 3 months.
Czarcasm, I’ve brought this to Cecil’s attention. He may amend the column, I can never predict how he’ll react. One of the joys of etymology is, of course, that words/phrases are in use before they’re written down, so there’s (almost) always a chance of finding an earlier mention.
I love Yogi Berra (I’m old enough to have seen him catch for the Yankees), and my favorite quote of his (as he arrived late for something): “This is the earliest I’ve ever been late.”
And the “It ain’t over till it’s over” quote was taken a step further by (excuse the expression) O.J. Simpson who said, “It ain’t over till it’s over, and even then it’s not over.”
Sorry everyone. No cite, just memory (faulty as it is). 
Q
Czarcasm’s cites are correct and the most up-to-date we have at this point.
I didn’t say all the things I said. - Yogi Berra
The very last episode of St. Elsewhere had a fat female opera singer, in full Wagnerian armored breastplate and helmet with horns, being treated at the hospital for laryngitis. Moments before the end of the episode, she burst forth into song.
I like this phrase even more!
My favorite (attributed) Yogi-ism is when he noticed his son watching a Steve McQueen movie on TV. “Huh,” said Yogi. “He must’ve made that one before he died.”
There was also a singing fat lady in the final episode of Cop Rock.
Back in 1985 the Kansas City Royals played the St. Louis Cardinals in the World Series. The Royals lost the first two games, at home no less, won one game then lost another. So the standing was that they were down 3-1. During the fourth game I still remember a picture in the newspaper, of a St. Louis fan holding up a sign that read “The fat lady is humming!”
Of course Kansas City went on to win three games straight and take the Series 4-3. To this day they remain the only team to lose the first two AT HOME and still go on to take the title.
Don’t know if that was directed at me:), but of course they’re correct. I was just veering off course as I am apt to do, Samclem. 
Quasi
As well as at the end of the Marx Brother’s Duck Soup, IIRC. EDIT: OK, she’s not actually that fat, now that I find the movie on youtube. But she definitely ends the show.
In 1986 the Mets lost the 1st two at home and won the Series.