Crawford was also perfect at the plate, going 4-for-4 and driving in a run as the AL champion Rays defeated the Red Sox for the third time in four games.
This year I've been trying to put pressure on (opposing teams)," Crawford told reporters after he became only the fourth Major League player since 1900 to steal six bases in a game.
"Last year was a little difficult for me (because of injuries)," the left fielder added. "But this year I definitely want that threat to kind of change the game a little bit."
He now has 318 career steals with a Major League-leading 17 coming this season.
"He's good. He's got the speed to do things like that," Boston catcher Jason Varitek told reporters.
James Shields (3-2) held the Sox to two runs and six hits over seven-plus innings for the victory. He struck out six and walked two.
"He always locates all his pitches," said Boston's Dustin Pedroia. "He keeps his team in the game."
Tampa Bay broke open a 1-1 game with two runs in the fourth inning on Jason Bartlett's run-scoring single and a fielder's choice by Michel Hernandez.
Crawford made it 4-1 when Red Sox pitcher Manny Delcarmen hit Pat Burrell with the bases loaded to force in a run.
Boston cut the lead to 4-3 on Kevin Youkilis's two-run homer in the eighth but Crawford's RBI single gave Tampa Bay another run in the bottom of the inning.
Brad Penny (2-1) pitched six innings for the Red Sox, giving up three runs and six hits. He struck out eight and walked two.
"I thought Brad really threw the ball," Boston manager Terry Francona said. "It was good, it was strong, it was down, it was over the plate."
