"There were big hits up and down the lineup," Yankees manager Joe Girardi said.
"I think we're going to be fine," Damon said.
Giambi, Cano and Betemit each began the day hitting well under .180. Damon has done better from the leadoff spot, and he homered for the Yankees' first run in the fourth inning and hit a go-ahead double in the fifth.
Giambi has spent the whole season searching for a groove. Along with the home run, the former AL MVP seemed equally pleased by a sharp lineout to center in his next at-bat.
"Hopefully, I'll get a few more to drop in," Giambi said.
Cleveland was trying to sweep the Yankees in a series of three or more games for the first time in nearly two decades. The last time the Indians did, they completed it by beating Tommy John on April 9, 1989 -- John, by the way, turns 65 in two weeks.
Mike Mussina (5-3) won his fourth straight start and earned his 255th career victory, holding on for five innings. Setup man Joba Chamberlain won his rematch with David Dellucci and Mariano Rivera closed for his ninth save.
The only drama after that came in the eighth when Chamberlain faced Dellucci. On Tuesday night, Dellucci rocked the hard-throwing reliever with a pinch-hit three-run homer.
Girardi later talked to Chamberlain about his penchant for shaking off his catcher's signs. This time, Chamberlain again shook off Jose Molina and then got Dellucci to swing over a slider for strike three.
Chamberlain, as usual, gave an exaggerated fist pump and hollered after fanning Dellucci to end the inning.
"That's who he is. He's not showing anyone up. He's going to show emotion," Girardi said. "He didn't look at Dellucci. He looked into our dugout."
That's not exactly how Dellucci saw Chamberlain's celebration.
"It is what it is. If he wants to yell and scream after a strikeout, I guess that's what gets him going," he said. "It's May baseball. The home run was in a much bigger situation. I didn't dance and scream.
"If a hitter did something like that, it would be bush. It's kind of interesting how a pitcher gets away with it," he said.
Said Chamberlain: "It didn't matter who it was, I just wanted to get them out."
Mussina moved into sole possession of 39th place on the career wins list, breaking a tie with Jack Morris and Red Faber.
Giambi's upper-deck drive landed just inside the right-field foul pole, with the star twisting his body and hoping to steer the shot clear. It was his sixth homer of the season, including two off Byrd on April 25 at Cleveland.
"I just cannot get Giambi out this year. I make the pitch, he hits it," Byrd said.
"He used to own me," Giambi said.
Damon blooped a two-out double in the fifth that drove in Cano, who opened the inning by hustling for a double.
The Indians had won five straight on the road. For the second straight game, slumping star Travis Hafner was benched. He's hitting only .209 with three home runs, and manager Eric Wedge is trying to find a way to get him to relax.
The game marked Cleveland's final scheduled appearance at Yankee Stadium. The Indians finished 262-456 overall at the ballpark, the Elias Sports Bureau said, starting with a 13-3 win on June 9, 1923, when Tris Speaker hit a grand slam.
Yankees slugger Hideki Matsui extended his hitting streak to a career-high 17 games. ... Cano and Betemit gave the Yankees their fourth set of back-to-back homers this season. ... Blake continues to deliver in the clutch. He is 14-for-33 with runners in scoring position, 7-for-67 otherwise. ... Rivera didn't record his ninth save last year until June 22. ... Indians outfielder Jason Michaels, who was designated for assignment Tuesday, was traded along with cash to Pittsburgh for a player to be named.