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: The A's are in split-squad action on Tuesday, with right-hander Dan Straily making his Cactus League debut against the Brewers at Phoenix Municipal Stadium in front of manager Bob Melvin. John Jaso will make his first start behind the plate since Wednesday, when he was hit by a pitch above his right elbow.
Lefty Tommy Milone gets the road assignment against the Cubs, along with batterymate Norris and Donaldson, Brandon Moss and Reddick.
Both contests are scheduled for 12:05 p.m. PT. Fans can listen to the home game free via an Athletics.com exclusive webcast.
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.