“Going…going…gone!” was the signature home run call of New York Yankees announcer Mel Allen (1913-1996). “Going, going, gone!” has been cited in print since at least 1780 and has been part of an auctioneer’s phraseology; “going” means that bids will soon be closed and “gone” means that the item is sold to the highest bidder. “Going, going, gone!” has been a baseball home run call since at least 1942.
 
The ball doesn’t always land in the stands for a home run. “Going…going…caught!” is the call of an embarrassed announcer who had thought that the ball would go for a home run.
 
“Going, going, gone, goodbye!” was the home run call of New York Mets broadcaster Ralph Kiner (1922-2014).
 
 
Wikipedia: Going, Going, Gone
Going, Going, Gone is a phrase used by auctioneers to indicate they are about to, and then accept, the final bid for an item in an auction, its also used to describe a home run in baseball as it sails through the air until it leaves the ballpark.
 
Wikipedia: Mel Allen
Mel Allen (Hebrew name: Mordechai ben Yehuda Elya; February 14, 1913 – June 16, 1996) was an American sportscaster, best known for his long tenure as the primary play-by-play announcer for the New York Yankees. During the peak of his career in the 1940s, 1950s and 1960s, Allen was arguably the most prominent member of his profession, his voice familiar to millions. Years after his death, he is still promoted as having been the “Voice of the New York Yankees.” In his later years, he gained a second professional life as the first host of This Week in Baseball.
(...)
Among Allen’s many catchphrases were “Hello there, everybody!” to start a game, “How a-bout that?!” or “Going, going, gone!” on home runs and “Three and two. What’ll he do?”
 
(Oxford Engish Dictionary)
go, v.
In auctioneers’ phraseology going! = on the point of being sold! gone! = sold!
1780 R. B. Sheridan School for Scandal iv. i. 44   I’ll knock ‘em down at forty pounds—Going—going—gone.
1814 Moore Sale of Tools ii,  Once, twice, going, going, thrice, gone!—it is yours, sir.
 
7 October 1942, Elyria (OH) Chronicle-Telegram, “Rustling Thru Sports” with Russ Davies, pg. 11, col. 1:
“Going, going, gone,” said the announcer and Kurowski trotted around the bases to follow Cooper across the plate.
 
13 May 1947, Traverse City (MI) Record-Eagle, pg. 10, col. 3:
HANK HITS: And It’s Going, Going, Gone!
(Photo—ed.)
EYES GO UP as Hank Greenberg hits one during a game withthe Phillies in Philadelphia and the eyes tell the story. The ball was good for a home run and gave the Pirates a win.
 
8 September 1952, Canton (OH) Repository, “Radio and Television: ‘Going, Going—Oops!’” by John Crosby, pg. 4, col. 3:
Mel Allen has returned to that old, old standby “Going, going, gone.” Mr. Allan (sic) has his own unique twist on this chestnut because frequently he calls out “going, going” and then is regretfully forced to explain that somebody caught it.
   
14 April 1957, New York (NY) Times, “Baseball Almanac"by John Lardner, pg. SM15:
JUNE 11
“It’s going, going—man, oh, man, he caught the ball, fans, in a truly great play!” shouts Mel Allen, popular broadcaster, as Hank Bauer of Yankees deposits medium-deep fly in hip pocket.
 
20 August 1961, New York (NY) Times, pg. SM40:
Sic Transistor
The sound of music—and of Mel Allen—is almost eerily pervasive in New York these summer days. The whine and thump of rock ‘n’ roll drowns out the thrushes in Central Park; the rapturous description of a “hi-i-igh fly-y-y ball—going, going” emanates from a baby carriage.
 
Google Books
Sports on New York Radio:
A play-by-play history
By David J. Halberstam
Lincolnwood, IL: Masters Press
1999
Pg. 110:
If Mel Allen’s classic home-run description was “going, going, gone,” Marty’s (Glickman—ed.) field-goal call was his stamp on the
broadcast. “It’s high enough, it’s deep enough, it’s through there, it’s through there!”
Pg. 353 (Ten Most Popular Phrases):
Mel Allen…..........“It’s going, going, gone!” signature home-run call