Manager Joe Maddon never gave serious thought to having Haynes bunt instead of go for the game-winner with two runners on.
"He told me if we got runners on first and second that he wasn't going to bunt me," Haynes said after his fourth career RBI ended the four-hour game. "In the back of your mind, you just want to make him right and go out there and get the job done."
Haynes lined an 0-1 pitch from Mike Timlin (2-2) to right field after Crawford, who was 3-for-6 and scored three runs, began the rally with a single and stole second. Timlin walked B.J. Upton before giving up the winning hit.
"Joe just took a chance and it paid off for him," Timlin said of the Rays' strategy.
The Red Sox stranded the potential go-ahead run in scoring position in the ninth, 10th and 11th innings. Scott Dohmann (1-0) got David Ortiz -- the only batter he faced -- to ground into an inning-ending double play with men at first and third in the top of the 11th.
Dan Wheeler worked out of a jam in the ninth, getting Ortiz to fly to center and striking out Manny Ramirez with runners at first and second. Kevin Youkilis and Jed Lowrie walked in the 10th before Julio Lugo grounded into a double play.
"We had our chances. We had the guys up that we wanted, but we weren't able to push a run across," Red Sox manager Terry Francona said. "Everybody gets frustrated. You want to win so bad. We got guys on, we couldn't push the run across."
The Rays rallied after falling behind knuckleballer Tim Wakefield 3-1, extending their season-best winning streak to four games. Their 12 wins overall match their most ever after 23 games.
Right-hander Matt Garza made his first start since leaving Tampa Bay's April 8 home opener with what the Rays described as radial nerve irritation in his pitching arm.
He retired the Red Sox in order in the first, then nearly didn't make it through a 40-pitch second in which he gave up a leadoff single to Ramirez before walking J.D. Drew and Sean Casey to load the bases with one out.
Casey was injured scoring on Ellsbury's hit, limping across the plate with a right hip flexor strain that forced him out of the game.
Wakefield was unbeaten at Tropicana Field until the Rays beat him here last Sept. 23. He's 19-3 all-time against Tampa Bay, including 9-1 on the road, but struggled after the Red sox gave him the early lead.
Crawford, who has a 12-game hitting streak, tripled and scored on Upton's sacrifice fly in the first and Tampa Bay trimmed Boston's lead to 3-2 on Evan Longoria's RBI single in the third.
Wakefield fell behind 4-3 in the fourth when Crawford tied it with an RBI triple and then scored on shortstop Lugo's fielding error. The Red Sox countered with a run in the sixth, then turned the game over to their bullpen.
"We got out of some hairy jams, based on either good pitches or good defense. They did the same," Maddon said. "It was really a well-fought game on both sides, and we just got the big hit out of Nate in the end."
Boston's Justin Pedroia singled in the 11th, giving him a 14-game hitting streak. ... Wakefield allowed four runs, six hits and five walks in six innings. Garza gave up three runs, five hits and walked three in five innings. ... Rays INF/OF Eric Hinske, who was with Boston last season, received his World Series ring before the game. ... Boston 3B Mike Lowell (sprained left thumb) began a rehab assignment with Triple-A Pawtucket. If all goes well this weekend, he could rejoin the Red Sox on Tuesday.