Kershaw, the reigning NL Cy Young Award winner with a new seven-year, $215 million contract, struck out two and walked one. He gave up three extra-base hits, including A.J. Pollock's RBI triple and Martin Prado's run-scoring double.
Yasiel Puig went 2 for 3 with an RBI double for the Dodgers.
Arizona's Brandon McCarthy allowed a run on four hits with two strikeouts and a walk in four innings.
The teams met for the first time since the Dodgers beat Arizona to clinch the NL West last year, annoying the Diamondbacks by celebrating in the Chase Field swimming pool.
"Obviously not very good," Kershaw said of his outing. "It wasn't very pretty. My body feels good, the arm feels good, it's just obviously there's a lot of stuff to work on."
Relievers J.J. Putz, Brad Zeigler, Addison Reed, Joe Thatcher and Josh Collmenter each threw a scoreless inning for the Diamondbacks.
Dodgers: Kershaw had trouble from the start, giving up a one-out single to Aaron Hill and an RBI double to Prado in the first inning. In the second, Gerardo Parra doubled with one out and took third on outfielder Joc Peterson's error. Pollock followed with a shot off the top of the glove of Carl Crawford in right field for an RBI triple.
"I wasn't throwing the ball where I wanted to," Kershaw said. "There's some off-speed pitches that I needed to throw better. ... There's just some balls that l left up."
Diamondbacks: McCarthy, who is coming off a difficult, injury-plagued season, was pleased with his start. He threw 52 pitches, 37 strikes.
Dodgers: Outfielder Matt Kemp, coming back from ankle surgery, is to have an MRI on Friday to assess the status of the healing process. He's not expected to be ready for the Australia trip.
Arizona: Outfielder Cody Ross is far ahead of schedule in his recovery from a gruesome hip injury he sustained while running the bases last season. He is cleared for virtually all spring training activity.
Neither Kershaw nor Dodgers manager Mattingly did anything to clear up speculation that the Dodgers' ace won't make the trip to Australia, where Los Angeles opens its season with two games against the Diamondbacks March 22-23. Asked if he wanted to go to Australia, Kershaw said, "ask Donnie."
Mattingly would say only that he's preparing four pitchers—Kershaw, Zack Greinke, Hyun-Jin Rhu and Dan Haren—and will make a decision later on who will start the Australia games.
Greinke already has said he has "zero" excitement about the Australia trip.
One of the big questions for Arizona this spring is how newcomer Mark Trumbo will do in left field.
He's certainly off to a good start, making a sliding, spectacular grab of a line drive off the bat of leadoff hitter Chone Figgins. Trumbo also made a nice lunging grab later in the game.
"I think there's a lot of skepticism," Trumbo said, "so the more I can produce and contribute, the more confidence people will have in me."