Despite a 44-39 record, New York pulled within 3½ games of Boston, the closest the Yankees have been to the AL East lead since the morning of April 23.
"We're clicking pretty good offensively, we're playing great defense, our pitching is really coming around," Giambi said. "Hopefully, we're all starting to come together."
Giambi also had an RBI single and raised his batting average to .274, the highest it's been since he went 1-for-2 in the season opener. He boosted his home run total to nine.
"I've worked hard to get to this point," he said. "I'm finally feeling good physically."
Giambi has four home runs, two doubles and a single in his last 10 at-bats, and has homered in three consecutive games for the first time since July 11-13, 2003.
"He's feeling pretty good about himself right now," Yankees manager Joe Torre said.
Cabrera became the first player to make his major-league debut as the Yankees' starting center fielder since Bernie Williams on July 7, 1991, according to the Elias Sports Bureau. Yankees manager Joe Torre was impressed with the way Cabrera called off second baseman Robinson Cano.
"He may have been pressing a little bit at the plate," Torre said, "but again, the most important thing for us is the defense he can give us in center field."
Cabrera went 1-for-4 with an opposite-field single to left in the fifth inning -- he plans to give the ball to his mother, Teresa.
"I feel really happy that I got my first hit here at Yankee Stadium," he said through a translator. "I felt no pressure. It was fun playing in front of 52,000 fans."
Mussina (9-5), who had three innings of fewer than 10 pitches, allowed two runs and six hits in seven innings, lowering his ERA to 3.97. Tom Gordon relieved in the eighth but left after three batters with tightness in his right shoulder. He is to have an MRI exam Friday.
"I'm not worried about it," Gordon said. "There's no pain at all, just some stiffness. I couldn't get loose the way I'd like to. I think I'm fine."
Aaron Boone returned to Yankee Stadium for the first time since 2003, when his 11th-inning homer off Boston's Tim Wakefield won Game 7 of the AL championship series. Boone, who received an ovation from fans, went 2-for-4 with two singles.
Kevin Millwood (3-7) gave up six runs and seven hits in six innings, dropping to 0-3 against the Yankees in his career during the regular season.
"He's had a lot of tough luck," Indians manager Eric Wedge said. "He's thrown the ball very consistently for us. We just haven't been able to give him a great deal of run support."
Millwood retired his first two batters with 0-2 counts and went 0-2 on Gary Sheffield, who singled to left. Rodriguez followed with a drive just over the right-field wall.
"When my hand hit the wall, I thought I had it," Blake said, adding jokingly: "It should have been fan interference."
Rob Marchese, a 41-year-old businessman from Queens sitting in a folding chair in the first row in the right-field stands, fumbled away both Rodriguez's and Giambi's homers.
"He was in the right spot tonight," Giambi said.
Grady Sizemore pulled Cleveland to 3-2 with a RBI grounder in the fifth, but New York broke open the game with a three-run sixth. Hideki Matsui had an RBI single, Millwood threw a run-scoring wild pitch when he slipped on the mound and Giambi hit his single.
Millwood, who made his Yankee Stadium debut, was impressed by the play of Cabrera, who is less than two weeks removed from Double-A.
"He looks like he's going to be a pretty good ballplayer," he said. "The hit he got was a mistake, and that's kind of the story of the night for me."
- Cleveland has lost nine of its last 11 games against New York and 10 of its last 11 at Yankee Stadium.
- To make room for Cabrera, the Yankees put RHP Carl Pavano (sore shoulder) on the 15-day DL retroactive to June 28.
- LHP Darrell May, acquired by New York last weekend from San Diego for reliever Paul Quantrill, is likely to be brought up to start Saturday.
- A moment of silence was observed before the game for victims of the London explosions.