"It's big," Aybar said of RBI No. 52 of a breakthrough season. "Right now we're trying to win as many games as we can. This was a big one for us."
It kept the surging Rangers 4 1/2 games off the Angels' American League West pace after Texas swept a doubleheader in Cleveland.
Kazmir met the challenge presented by Felix Hernandez, one of the game's dominant pitchers, with seven strong innings after yielding a homer to Franklin Gutierrez, the second man he faced.
"Throw up zeros and get as deep as I can," Kazmir said, defining his objective. Mission accomplished.
Kazmir's first win in his new No. 22 disappeared with Sweeney's drive over the center-field wall, beyond even Torii Hunter's storied reach.
"We felt good with the way we got there," Angels manager Mike Scioscia said, "with the way Kazmir threw the ball, the way Kevin threw the ball.
"Brian's been doing a job for us. Unfortunately, he didn't get a fastball where he wanted to Mike Sweeney, and we were in a tie ballgame."
Courtesy of Aybar, Abreu and Willits' collaborative effort, the win went to Matt Palmer (10-1) after the fifth Angels pitcher escaped a two-on jam in the top of the 10th.
Coming back from a shoulder issue that had sidelined him for nine games, Jason Bulger left two runners in scoring position in the ninth, striking out Ken Griffey Jr. and retiring Mike Carp following Sweeney's tying blow.
"I kept it down," Bulger said of his big curveball that struck out Griffey, the legend, with runners at second and third and one out. "You don't want to leave it hanging.
"I've faced him a couple times. That awe -- no matter who's in the batter's box -- is gone."
Kazmir certainly didn't hesitate in coming right after Ichiro Suzuki, catching the great one looking at a fastball to open the game and retiring him on a grounder next time.
Ichiro reached on Chone Figgins' second error in the sixth, but was erased on Gutierrez's double-play grounder to the third baseman. Ichiro grounded out his last two times at the plate, ending a rare 0-for-5 night.
That kept Ichiro at 195 hits in pursuit of his ninth consecutive 200-hit season.
"That shows you what kind of pitcher Kazmir is," Mariners manager Don Wakamatsu said. "Ichiro usually handles lefties pretty well, but it looked like [Kazmir] made him uncomfortable."
The 25-year-old Houston native came right after Ichiro with his premium stuff.
"Attack the zone and try to expand it," Kazmir said of his approach. "Try to get him on his heels, in and out. You have to throw quality pitches."
Kazmir used the ugliest pitch of his night, a hanging slider in Rob Johnson's eyes, to strike out the catcher and finish the seventh. This left Bill Hall at second after he'd singled and advanced on a passed ball by Mike Napoli.
"That was a designed hanging slider," Kazmir said, grinning.
Batista (6-4) absorbed the loss after Hernandez held the Angels to two runs, one earned, across seven innings.
Composing himself after Gutierrez's jolt, Kazmir found his rhythm in the rematch of Hernandez's 3-0 decision six days earlier in Seattle.
Kevin Jepsen's scoreless eighth had Fuentes in position for his 40th save. But Sweeney lifted the lefty's second pitch over the wall in center for his seventh homer of the year, depriving Kazmir of his first win as an Angel. He's 8-8 overall.
Sweeney wasn't thinking about going deep.
"Nah, I have never actually tried to hit a home run and actually done it," he said. "At the moment, it gave us a chance to play more baseball, but here we are at 10:30 [p.m. PT] and it doesn't mean squat.
"We're down to the point of the season where we pretty much have to win every game."
After Gutierrez's third career homer against Kazmir in his 13th at-bat, the Angels tied it in the third without a hit against Hernandez. He walked three men before Abreu delivered a sacrifice fly to score Napoli.
Hernandez's control cost him again in the fifth when Figgins worked a one-out walk, advanced on Maicer Izturis' single and scored when shortstop Jack Wilson was unable to handle Abreu's line shot on one hop. Wilson was charged with an error, making Figgins' league-high 104th run unearned.