Rays right-hander Alex Cobb went five innings, giving up four runs on 11 hits with one walk. Neither starter got a decision.
The Red Sox responded with two runs in the bottom of the inning. Andrew Benintendi and Mookie Betts hit back-to-back doubles for one run, and Mitch Moreland singled in another.
A Corey Dickerson homer in the fourth extended the Rays' lead to 4-2, but Red Sox slugger Pablo Sandoval bashed a game-tying homer in the bottom of the frame.
Tim Beckham led off the fifth with his first home run to push the Rays ahead, but the Red Sox got to reliever Danny Farquhar in the seventh when a Moreland single scored two.