Mike Lowell homered to tie the score against Devil Rays closer Al Reyes (1-2). Jason Varitek had a ground-rule double an out later and Crisp followed with a hit into short right to bring in the game-winning run.
Eric Gagne (3-0) pitched a scoreless ninth with three strikeouts to earn the win, his first in a Red Sox uniform. Jon Lester, making his first home start since returning to the team after beating lymphoma, allowed two hits and a run with four strikeouts and one walk in seven innings.
Dustin Pedroia had two hits for the Red Sox, who have won three of their last four games. Boston increased its lead over New York in the AL East to five games.
Tampa starter Scott Kazmir pitched six innings of four-hit, shutout ball with eight strikeouts and three walks.