Craig Breslow (1-0) picked up the win with 1 2/3 innings in relief of Jake Peavy and Koji Uehara followed up with a four-out save that gave Boston the win in Game 4 of their American League Division Series and a date with either Detroit or Oakland in the next round.
Xander Bogaerts scored on a wild pitch and Shane Victorino drove in a run in a two-run seventh that put Boston ahead 2-1. The Red Sox added a run in the ninth on a sacrifice fly by Dustin Pedroia that brought Bogaerts in from third.
Jake McGee (0-1), the fifth of nine pitchers deployed by the Rays, suffered the loss after allowing two runs on one hit.
Tampa Bay took a 1-0 lead in the sixth when Yunel Escobar led off with a double off of Peavy, moved to third on a ground out and then scored when David DeJesus singled. Peavy retired the next hitter and then gave way to Breslow, who fanned James Loney to put an end to the uprising.