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Teixeira's three-run shot and Martin's two-out single in the eighth inning overcame the Angels' 5-2 edge. The Yankees, with the best record in the majors, returned from the All-Star break and won for the fifth time in six games.
"It's a new start," Teixeira said. "The first half really doesn't mean anything."
Trumbo homered in his fifth straight game against the Yankees. His three-run shot in the seventh made him the first Yankees opponent to homer five games in a row since Albert Belle in 1994, STATS LLC said. Harold Baines in 1984-85 is the only other player to do that against New York, the Yankees said in citing the Elias Sports Bureau.
"Yeah? I guess, pretty cool," Trumbo said. "But if it wasn't enough to get a win, it's just a novelty."
Trumbo nearly hit another homer, but his bid for a two-run drive to right field was caught by a leaping Swisher at the top of the wall in the eighth. Trumbo is 11 for 22 with five homers and 10 RBI against New York this season.
"If he doesn't make that catch, we probably don't get back in the game," Teixeira said.
The Angels had two runners caught stealing, then Howie Kendrick was thrown out at second base to end the game. He tried to advance when a pitch from Rafael Soriano bounced away from Martin, but the catcher made a strong peg for the final out.
The Yankees broke loose after C.J. Wilson was pulled following a sharp seven-inning start.
Teixeira's tying shot was his 17th home run -- he has 36 career multihomer games, three this season.
Downs retired the next two batters and left after walking Swisher. Kevin Jepsen relieved and, after pinch-runner Dewayne Wise stole second and pinch-hitter Raul Ibanez was intentionally walked, Martin sliced a single to right.
Martin ended an 0-for-30 slump last Saturday in his last game before the All-Star break.
"I think it's important for him," Yankees manager Joe Girardi said. "I always talk about how players want to feel they're contributing and he had a great night tonight.
"Threw three guys out, the game-winning hit," he said.
Soriano wound up with his 21st save in 22 chances. Chad Qualls (1-0) got his first win with the Yankees.
Trumbo was picked by Yankees star Robinson Cano for the home run derby in Kansas City this week and finished among the leaders. The Angels slugger did equally fine on a humid evening in the Bronx, sending a shot into the left-center field bleachers against Hiroki Kuroda for his 23rd homer and a 4-2 lead.
Albert Pujols singled to start the Angels' seventh-inning rally, then hit a ground-rule double in the eighth that made it 5-2. He stumbled coming around first base and hobbled into second, prompting Angels manager Mike Scioscia and a trainer to check on him. Pujols stayed in the game.
Scioscia said Pujols earlier fouled a ball off his foot and "felt just a little jolt or something" coming around the bag.
"He should be OK," Scioscia said.
Angels rookie Mike Trout, leading the league in hitting and steals, doubled and singled, stole two bases and scored a run.
A prep star in New Jersey who worked out for scouts at the old and new Yankee Stadium, Trout's favorite player growing up was Jeter. They ate lunch together at the All-Star game, and Jeter made Trout's family and friends feel welcome in the 20-year-old's first game at the ballpark -- the 20-plus members of the Trout rooting section sat in Jeter's suite.
Erick Aybar also homered for the Angels.
- Jeter has scored 1,818 runs, two more than Carl Yastrzemski for 16th place on the career list.
- Cano extended his hitting streak to a season-high 16 games.
- Yankees LHP CC Sabathia (strained left groin) threw a simulated game. If he's OK, he'll start Tuesday night vs. Toronto. Asked his goal for the second half, the ace said: "Stay healthy."
- Yankees LHP Andy Pettitte (broken left leg) rejoined the team after recuperating at home in Texas. He's targeting Sept. 1 to be back in the rotation.
- Yankees OF Brett Gardner (right elbow) is ready to swing in a simulated game Sunday.
- New Brooklyn Nets star Joe Johnson, who said he didn't play baseball growing up, threw out the first ball and tossed a soft strike. Nets coach Avery Johnson led the Bleacher Creatures in the roll call.