Forsythe started the winning rally with a one-out walk. Evan Longoria had three hits and an RBI.
Rays prospect Richie Shaffer went 0-for-2 with a hit-by-pitch and a walk in four plate appearances before leaving the game in the seventh.
Jake McGee (1-1) pitched a perfect eighth to get the win and Brad Boxberger worked out of a jam in the ninth for his 27th save. David Robertson (4-3) took the loss.
Jose Abreu and Carlos Sanchez each had a two-run home run for Chicago, which has dropped four of five following a hot streak.