But Bay connected, clearing the 37-foot-high Green Monster above the 379-foot marker, and Youkilis hit a solo shot in the 11th to give the Red Sox a 5-4 win over the New York Yankees.
It was Boston's eight consecutive victory.
"It's tough to close out a game in this ballpark," said Yankees outfielder Johnny Damon, who used to play for the Red Sox. "In this ballpark, one swing of the bat can get you back in the game."
Ramon Ramirez (2-0) allowed one hit and one walk in one inning for the win.
Damaso Marte (0-1) struck out three and allowed just one hit: Youkilis' fifth homer of the year and the second game-ender of his career.
"It was a great night," Youkilis said. "Back and forth the whole game."
Rivera, who hadn't been called upon to get more than three outs in his first seven appearances this season, came on with two outs in the eighth and struck out Dustin Pedroia with runners on first and second to end the inning. In the ninth, he gave up a one-out single to Youkilis and, one out later, Bay's homer to tie it.
"When you see him come in, it's not a confidence booster," Red Sox manager Terry Francona said. "We have put a dent in him more than anybody else - but we've also seen him more than anybody else."
After Mike Lowell singled, Rivera struck out Jason Varitek to send it to extra innings.
"Mo got him out and then was one out away from closing out this game," Yankees manager Joe Girardi said. "It's unfortunate, but it's still one game."
Jonathan Papelbon stranded two in the top of the 10th. Ramirez walked Jorge Posada to lead off the 11th and Robinson Cano singled to put runners on first and second with one out, but Melky Cabrera grounded into a double play to end the threat.
Marte struck out David Ortiz to start the bottom of the 11th - his fourth whiff of the night - but Youkilis hit a 2-2 pitch to end it.
Joba Chamberlain held on through 5 1-3 innings thanks to four double plays. Boston finally chased him in the sixth, tying it 2-2 with three straight hits before Phil Coke got a pair of popups to end the inning.
Mark Teixeira, who snubbed the Red Sox and signed with their rivals, broke a seventh-inning tie in his first appearance at Fenway Park in a Yankees uniform. Teixeira, one of the pieces of New York's $423.5 million offseason spending spree, blooped one into center to break the tie, then New York took a 4-2 lead on Robinson Cano's sacrifice fly.
Notes: Arthur Giddon, a former bat boy for the Boston Braves, will be the honorary bat boy for the Red Sox on Saturday - his 100th birthday. ... Jon Lester pitched six innings, allowing two runs on seven hits and three walks while striking out seven. Chamberlain gave up one earned run and two in all, on nine hits and four walks while striking out two. He also had a balk and a throwing error. ... Teixeira's third-inning single was the 1,000th of his career. ... Cody Ransom was pulled for a pinch-runner after coming up lame taking third on a groundout.
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