Kershaw (13-10) scattered four hits, walking none and striking out nine. The only run he yielded came on Tony Abreu's ninth-inning RBI double.
Kenley Jansen came on for Kershaw with no outs and a runner on second in the ninth, and shut down the Diamondbacks for his second save.
Oeltjen scored A.J. Ellis with a triple in the third inning to break a scoreless tie and the Dodgers went on to win the first contest in a three-game series at Chase Field.
Jay Gibbons drove in a run with a sacrifice fly and Reed Johnson added a solo homer for Los Angeles.
Barry Enright (6-6) took the hard-luck loss for the Diamondbacks, giving up a pair of runs on five hits over eight innings.