Braden (7-7), reinstated from the bereavement list before the game to pitch for the first time since July 1, allowed five hits, two walks and struck out six. The left-hander, who had spent most of the past week with his ailing grandmother, is just 3-2 despite allowing two earned runs or less in each of his last eight starts.
"He looked a little fatigued at the beginning," A's manager Bob Geren said. "He had a long week and he came out and the game was his."
Braden acknowledged he "felt bad" early on.
"We won. That's the most important part," Braden said. "Was I happy with tonight? Not really, but I got through it."
Braden arrived in Florida around 11 p.m. Friday.
"When you don't sleep for four days and don't eat anything, it'll be brutal," Braden said. "I didn't pick up a ball until two days ago."
Garza allowed three runs and seven hits in 6 1/3 innings as Tampa Bay's nine-game home winning streak ended.
"I made one mistake and paid for it," Garza said.
"It just got away from us," Tampa Bay manager Joe Maddon said. "It shouldn't, but it did."
Davis replaced right fielder Ryan Sweeney in third.
"He popped a ball up and I didn't like the way he ran down the line," Geren said of Sweeney. "Sometimes players take their frustrations out different ways. He didn't run down the line. He's never made that mistake before and I'm sure after today we won't again."
Oakland loaded the bases with two outs in the first, but failed to score when Garza retired Kurt Suzuki for his third strikeout of the inning. The right-hander struck out seven and walked one.
Athletics first baseman Jason Giambi left in the fifth with a bruised right elbow, and is day to day. He was hit on the elbow by a pitch during the second.
Navarro was scheduled to undergo tests at a hospital to see if he has a concussion.
"He was woozy and not feeling well," Maddon said.
Hernandez was first charged with a catcher's balk, but that ruling was changed by the official scorer to an error after the game.
Maddon, after watching a replay, said Nelson's call that let a baserunner advance one base on the play was correct.
Maddon said his post-All-Star game rotation will be: RHP James Shields, LHP Scott Kazmir, Garza, LHP David Price and RHP Jeff Niemann. ... Oakland manager Bob Geren said Nomar Garciaparra will start playing around one game a series at first base. The move will allow Giambi to get additional time off. ... Tampa Bay C Shawn Riggans (right shoulder) could play in a minor league in the next few days. ... To make room on the roster for Braden, RHP Jeff Gray was optioned to Triple-A Sacramento.