The favourite was in the middle of the pack for most of the 1-1/16 mile race for male two-year-olds before he ran down the leaders in the final furlong on an overcast day in Kentucky.
Game Winner, who was an even money bet at post time, was followed by 40-1 Knicks Go, who finished second, and 67-1 Signalman, who came in third.
It was not a pretty victory as the chestnut colt, ridden by jockey Joel Rosario, did not break well from the gate and bumped into Knicks Go while making his move from the outside down the stretch.
"You talk about a rough trip," said Baffert, who trained this year's Triple Crown winner Justify as well as 2015 Triple Crown winner American Pharoah.
"Joel was a little patient back there," he said of his jockey, who had to battle back from seven lengths off the lead to secure Baffert's 15th career Breeders' Cup win.
"I felt a lot of pressure coming in here because I was really looking forward to this colt.
"I know he is a really good horse and he had to show it today that he was the best," he said.
Game Winner will hope to follow in the footsteps of Juvenile winners Nyquist in 2015 and Street Sense in 2006, who went on to win the Kentucky Derby at Churchill Downs the next year.
"He likes Churchill Downs so that's a good sign," Baffert said with a laugh.
"We've just got to keep him healthy," he said of the horse, who improved to 4-0 in his young career.
"Thank the lord that this horse came through and everything went well for us today because these races are hard to win."
Rosario, who won the Juvenile Fillies on the back of Jaywalk earlier in the day, will look to continue his winning ways when he rides favourite Accelerate in the $6 million Breeders Cup Classic on Saturday.
Baffert will look to do the same as two of his horses, McKinzie and West Coast, will compete in the showcase race. (Reporting by Rory Carroll, editing by Ed Osmond)
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