The Indians stopped Tanaka's bid to become the major league's first 13-game winner. The rookie All-Star right-hander (12-4) allowed season worsts of five runs and 10 hits in 6 2-3 innings.
But the night belonged to Brantley, who hit a leadoff homer in the seventh and had RBI doubles in the first and fifth, raising his average to .328. He also ran down Derek Jeter's fly ball to deep center on the warning track in the first when New York scored twice.
''He's a pretty good player, and he's getting better,'' Indians manager Terry Francona said with a smile.
Brantley has 14 homers and 60 RBIs, both team highs. He also is tied for the AL lead with 10 outfield assists.
''He's a young player that's really developed into one of the better outfielders in the game all-around,'' Yankees manager Joe Girardi said. ''And he showed why tonight. It's why he's an All-Star.''
Brantley chose to focus on the win more than his personal performance.
''We did a great job as a team having quality at-bats all night,'' he said. ''I was trying to get mistake pitches. I was lucky enough to get a couple of balls out over the plate and get good swings on them.''
Tanaka, who lost for the third time in four starts, took a 3-2 lead into the sixth before Swisher, a former Yankee, put Cleveland on top.
''It was the first time we've ever seen him,'' Swisher said. ''The first couple of at-bats you're kind of feeling him out. In that situation from the scouting reports and video we've seen, he loves that split-slider. When I got in that two-strike count, he'd been burying that split all day long. For me, I was just lucky enough for him to hang that one.''
Lonnie Chisenhall started the sixth with a single. Swisher then homered on a 1-2 pitch.
Trevor Bauer (3-4) allowed three runs in seven innings and retired 13 of the last 14 hitters he faced. New York's only baserunner in that stretch came on Swisher's fifth-inning error. Bauer struck out six and walked two in winning for the first time since June 16.
Cody Allen pitched the ninth for his 10th save.
The Yankees were held to four hits, none after Brian McCann's one-out single in the third.
The Indians committed three errors, giving them a major league-high 75 for the season.
Swisher heard boos from the crowd earlier in the game after going 0 for 2 and having a routine groundball go through his legs.
Tanaka allowed a run in a 25-pitch first inning and gave up two hits in the second. He retired seven straight before allowing a run in the fifth when Brantley had a two-out double after Chris Dickerson led off with a single.
Tanaka struck out five and walked one.
''He did get on a roll in innings three, four and five and then the mistakes kind of caught up to him,'' Girardi said. ''It happens, we know he's not going to be perfect, we understand that. We know how good he's been but we know he's not going to be perfect.''
Jason Kipnis started the first with a single, stole second and scored when Brantley doubled to right.
Gardner walked to start the game and moved to second on Jacoby Ellsbury's single. Dickerson attempted a diving catch of Teixeria's bloop hit, but the ball deflected off his glove and rolled toward center, allowing Gardner to score.
New York pulled a double steal when McCann struck out swinging. Gomes' throw to second went into center field and Ellsbury scored. Teixeria's stolen base was his first of the season.
RHP Brandon McCarthy (3-10) will make his Yankees debut Wednesday night. He was acquired from Arizona on Sunday. ''It's the Yankees,'' he said. ''It's a team that's in contention. It's a team that's known for winning, there's really no other answer than it's just the Yankees.'' ... The Indians recalled C Roberto Perez and LHP Nick Hagadone from Triple-A Columbus. ... Yankees OF Carlos Beltran (soreness behind right knee) was out of the lineup for the second straight game. ... Girardi said RHP Shane Greene, who recorded his first major league win Monday, will start Saturday against Baltimore. ... RHP Josh Tomlin (5-6) starts for the Indians on Wednesday.