The 4/5 favourite was in the middle of the talented pack for most of the 1-1/16 mile race for male 2 year-olds before he ran down the leaders down the stretch on an overcast day in Kentucky.
The 30-1 Knicks Go finished second and 54-1 Signalman was third.
It was not a pretty victory as Game Winner, ridden by jockey Joel Rosario, 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.
"I felt a lot of pressure coming in here because I was really looking forward to this colt. He had to show it today that he was the best.
"He likes Churchill Downs, so that's a good sign."
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 the next year.
(Reporting by Rory Carroll, editing by Ed Osmond)
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