It was Boston's second straight last at-bat win after losing its previous five games.
J.D. Drew hit a grand slam and Darnell McDonald homered in his second straight game for Boston. Mike Lowell also connected.
Josh Hamilton hit a three-run homer in the Texas seventh that made it 7-all.
Neither team did much in the extra innings until Marco Scutaro singled with one out in the 12th and alertly tagged up on a medium deep fly out to center, sliding in barely under second baseman Joaquin Arias' tag. After Dustin Pedroia was intentionally walked, Youkilis delivered against Dustin Nippert (1-2).
Texas was successful on all three steal attempts after swiping a team-record nine on Tuesday night. The Rangers are 30-for-30 over the past two seasons against the Red Sox. Boston's catchers have thrown out just one of 34 runners overall who have tried to steal this year.
Hideki Okajima (2-1) pitched one hitless inning for the win.
A big cheer went up from the Fenway Park crowd that was left in the middle of Michael Young's at-bat when the Boston Bruins won their NHL playoff game in double overtime. The Red Sox wrapped up their extra-inning win about eight minutes later.
McDonald was selected from Triple-A Pawtucket on Tuesday and became an instant hit that night. He tied the game with a pinch-hit, two-run homer in the eighth and won it in the ninth with a bases-loaded single off the left-field wall.
Down 7-4 in the seventh against Boston ace Josh Beckett, Hamilton homered deep into the center-field bleachers following two walks.
Beckett struggled with his control and was tagged for seven runs — six earned — on six hits in seven innings. He walked five and stuck out four.
Texas led 4-0 after two innings. The Rangers also wasted a 6-1 lead in Tuesday's loss.
Boston moved ahead 5-4 in the third when Drew lofted a fly ball down the right-field line that hooked around the foul pole into the second row of seats for his fifth career slam. McDonald, Bill Hall and Scutaro each singled before Drew's homer.
McDonald homered into a camera area in center, making it 6-4 in the fourth.
Lowell, starting at DH, homered into the Green Monster seats.
Rangers starter Matt Harrison gave up six runs in four-plus innings.
Notes
Texas RF Nelson Cruz left the game with fatigue in his right hamstring after leaving Tuesday's tilt with a cramp in the same area. … Red Sox manager Terry Francona had a meeting with struggling David Ortiz a few hours before the game after deciding to pinch hit for him Tuesday and using Lowell as the DH on Wednesday. "He's been an unbelievable player here," Francona said. "I just wanted to let him know that we care about all our players and that we try to do what's right for our team." … Harrison had allowed just one earned run while going at least six innings in his first two starts this season. … Boston's Daisuke Matsuzaka pitched 5 2/3 innings, allowing three earned runs on four hits, striking out eight and walking none in what is expected to be his final of three rehab starts for Triple-A Pawtucket at Lehigh Valley on Wednesday. … Rangers reliever Neftali Feliz was clocked at 101 on consecutive pitches on the stadium radar gun.