"It's phenomenal," said Dallas Braden, who pitched five scoreless innings to get the win.
"It was pretty much timing," Sweeney said. "I just jumped, got lucky and caught it. ... He hit it dead center. I think if he would have pulled it a little bit, it might have gone."
Kinsler, already with seven homers, slammed his helmet to the ground with both hands when he realized Sweeney caught the ball.
"It was a good catch. There's nothing more I can say," Kinsler said. "I was hoping it would be on the grass and out of his reach."
A's manager Bob Geren said the 6-foot-4 Sweeney "needed every inch" to make the play.
Braden (3-2) allowed five hits and walked four, but managed to work out of trouble. The left-hander struck out six, escaping bases-loaded jams his last two innings with Ks.
Before Matt Holliday homered in the ninth, Oakland got runs on two sacrifice flies and a bases-loaded walk issued by Eddie Guardado, who threw four balls to the only batter he faced and was ejected by umpire Angel Campos after he had already been taken out of the game.
Holliday's homer was his first in 18 games since being acquired from Colorado over the winter.
Andruw Jones homered for Texas in the ninth, right after Michael Young was thrown out trying for a triple on a ball hit off the left-field wall. Michael Wuertz, closing with Brad Ziegler out sick, still got his first save.
Padilla was pulled after loading the bases with no outs in the seventh, including a close call when the right-hander was covering first and stepped on the base simultaneously with Sweeney, who was called safe.
Guardado was pulled after walking Jack Cust and while going to the dugout had words with Campos, who remained near home plate and ejected the pitcher.
The reliever then slung his glove and cap toward the stands near the Rangers dugout. Guardado, who has a 9.64 ERA in nine appearances, raised both arms over his head to incite the fans, who responded with cheers.
"It was all my fault. I was just letting out some frustration, and he happened to be standing there," Guardado said. "I've not been doing my job. When that happens, it builds up."
Kurt Suzuki added a sac fly for a 3-0 lead.
Two of Oakland's three shutout losses are in games started by Braden, and it was a scoreless game when he threw the last of his 111 pitches.
Powell started at first base and Bobby Crosby moved to third when Eric Chavez was scratched. Geren said Chavez, who has played in only three of the last 13 games, aggravated his sore right elbow during batting practice.
Rangers C Jarrod Saltalamacchia left after two innings because of eye irritation. ... A's RHP Justin Duchscherer has started a throwing program in Arizona in his comeback from right elbow surgery. He is at least a month away from pitching in a rehab assignment. ... Rangers RHP Kris Benson, on the DL with tendinitis in his right elbow, threw on the side without problems and will make a rehab start Sunday at Double-A Frisco.