Kershaw, who was also hit hard in his first spring start, was expected to go three innings on the day, but the A's chased him out of the game before he recorded an out in the third. And they still weren't done.
Facing right-hander Brandon League, Josh Donaldson walked and scored on Josh Reddick's two-run double. Two more walks later, Norris collected a two-run single.
The A's sent 11 men to the plate in the inning, wiping away a two-run lead gained by the Dodgers against A.J. Griffin.
Making his Cactus League debut, Griffin struck out two in a scoreless first inning but allowed three hits, including a two-run triple to Dee Gordon, in the second.
Lefty Sean Doolittle struck out two in one perfect inning, and closer Jim Johnson followed with a 1-2-3 inning of his own.
Andre Ethier and Miguel Olivo led the Dodgers' offense with two hits apiece, and Rule 5 bullpen hopeful Seth Rosin had another impressive outing, keeping the A's off the board on three hits in three innings.
Up next: With right-hander Zack Greinke injured and unable to pitch, the Dodgers will use all relievers on Tuesday and start former All-Star closer Brian Wilson against the Mariners in a game available live on MLB.TV at 12:05 p.m. PT. Outfielder Carl Crawford, who hasn't played since straining his right quad on Wednesday, may return to the lineup.
Jane Lee is a reporter for MLB.com. Read her blog, Major Lee-ague, and follow her on Twitter @JaneMLB. This story was not subject to the approval of Major League Baseball or its clubs.