The right-hander gave up five hits and three runs in 5 1-3 innings. He struck out four, walked two and hit a pair of batters, leading to the A's first two runs.
Oakland started its scoring surge after home plate umpire Paul Nauert ruled Craig Gentry was hit by a pitch on his hand leading off the fifth. Giants manager Bruce Bochy challenged the play, which was upheld after a video review.
Gentry stole second, advanced to third on Coco Crisp's bunt single and scored on John Jaso's groundout for the A's first run.
Josh Donaldson also was hit by a pitch by Vogelsong on his left hand leading off the sixth. Nauert originally called it a strike before seeing Donaldson's hand swell.
Then Nauert asked first base umpire Angel Hernandez if Donaldson, who said his hand was just sore afterward, swung at the pitch. Hernandez signaled no, and Bochy came out and had a heated conversation with Nauert after the call, wondering how it could be reversed.
"Then it got ugly," Bochy said.
Callaspo came through later in the inning with a two-run double off Juan Gutierrez to extend Oakland's lead to 3-0. Brandon Moss and Donaldson each drove in a run off Gutierrez in the seventh after Giants shortstop Brandon Crawford committed two throwing errors.
The teams will play in Oakland again on Tuesday night before the series shifts to San Francisco for the final two games.
NOTES: Giants 3B Pablo Sandoval, who left Saturday's game because of a bruised left elbow after he was hit while swinging at a pitch, sat out for the second straight game. ... Athletics RF Josh Reddick took batting practice and fielded fly balls for the first time since re-aggravating his right knee injury June 28 at Miami. ... Giants 2B Marco Scutaro (back) went 1 for 3 with an RBI single in his first rehab assignment for Triple-A Fresno on Monday night at Sacramento. ... Giants CF Angel Pagan was sent to Arizona to resume his rehabilitation from a nagging back injury. ... Giants lefty Madison Bumgarner (9-6, 3.09 ERA) takes the mound opposite Oakland's Sonny Gray (8-3, 3.08) on Tuesday.