Yoenis Cespedes and Brandon Inge both singled before Norris crushed a 3-2 fastball into the seats in left-center field. Norris' first-career homer helped the A's avoid a three-game sweep in the Bay Bridge series.
Jimmy Miller (2-0) earned the win after throwing 1 1/3 shutout innings. A.J. Griffin, who made his major league debut, gave up two runs on three hits and struck out four over six-plus innings. Griffin retired 13 straight after giving up a two-out hit in the second.
Coco Crisp doubled and scored and Josh Reddick drove in a for the A's, who have won nine of their last 12.
Santiago Casilla (1-3) was pegged with the loss after giving up the Norris homer. Matt Cain started for the Giants and allowed one run on three hits and struck out seven.
Buster Posey hit a two-run homer and Ryan Theriot had two hits and scored a run for San Francisco, which has lost five of eight.
The Giants took a 2-0 lead in the first when Posey launched a two-run homer. Theriot singled before Posey hit his ninth four bagger of the year.
But the A's came back with a run of their own in the home half to make it 2-1. Crisp doubled then scored when Reddick singled to right.
With two runners on in the sixth Seth Smith fouled one down line in right. The ball clearly hit the foul line but was ruled foul. The ball rolled into the corner and likely would have cleared the bases and given Oakland the lead.
Oakland wasted an opportunity to score in the eighth. Pinch-hitter Brandon Hicks doubled then moved to third on Crisp's sacrifice bunt. Jemile Weeks worked a walk to put runners on the corners with one out, but Reddick grounded into an inning-ending double-play.
Oakland's starting pitchers have allowed four hits or fewer in each of its last eight games...The A's have won nine of their last 11 interleague games...Cain hasn't lost since May 1. He won eight straight starts from May 12-June 18...No San Francisco pitcher has ever won nine consecutive starts.
For more on the A's, check out Athletics Nation.