“Fall classic” is another name for baseball’s World Series that takes place in October. The name “fall classic” has been cited in print since at least July 1914.
 
   
Wikipedia: World Series
The World Series is the annual championship series of Major League Baseball, played between the American League and National League champions since 1903. The winner of the World Series championship is determined through a best-of-seven playoff and awarded the Commissioner’s Trophy. Because the World Series is played in October, it is sometimes called the Fall Classic.
 
Chronicling America
30 July 1910, El Paso (TX) Herald, “Turf Battle In Florida” by Burt E. Collyer, pg. 3, col. 3:
In that event nearly all the fall classics for three-year-olds in the middle west are at the mercy of the Fizer crack.
 
Chronicling America
13 July 1914, Washington (DC) Herald, pg. 8, col. 3:
No one outside of Chicago fandom would care particularly to see the fall classic confined to the Windy City as it was in 1906, and of which there is a slight possibility just now.
 
27 July 1914, Washington (DC) Post, “Stanchest Foes of A. L. Leaders” by Frank Chance (Manager of the Yankees),  pg. 8, col. 3:
At least there is hope ahead for the five clubs which have fought so valiantly to determine a new American League entry in the annual fall classic.
 
3 October 1915, Macon (GA) Telegraph, “Jack Barry, With Red Sox Now, Holds Record For Taking World Series Coin,” pg. 10:
Barry’s fifth will come with the 1915 series, as the Red Sox will form 50 per cent of the fall classic.
 
LA84 Digital Archive
21 October 1916, Sporting Life, “Changes in the World’s Series” by W. J. McBeth, pg. 10, col. 3:
“Far better no World’s Series at all”—and there was no doubt of Johnson’s earnestness—“than that fandom should be plucked or conditions tolerated that permitted the most humble bleacher patron to feel that he was being discriminated against in the annual Fall classic.”
   
Google News Archive
13 April 1918, Windsor (Ontario) Record, pg. 9, col. 4:
BASEBALL SHARP
PICKS REDS AND
SOX AS WINNERS
By H. C. HAMILTON
(...)
On just one thing hinges the chances for the Reds to kick history and precedent into a cocked hat and get into a world’s series for the first time in the history of the fall classic.
 
Google Books
2 November 1918, Advertising & Selling, “Who Will Win?” by H. E. Provest, pg. 16, cols. 2-3:
In base ball there is always a world’s series and more or less of an uncertainty as to who will collect the big end. Good management and team play here often is a deciding factor in the fall classic.
 
Google Books
The Dickson Baseball Dictionary
By Paul Dickson and Skip McAfee
New York, NY: W.W. Norton & Co.
2009
Pg. 303:
Fall Classic Syn. of World Series. The term is one o baseball’s oldest cliches, fully expressed in the notion of “titans clashing in the Fall Classic.” Sometimes lower-cased as “fall classic.”