Kershaw (3-1), the reigning National League Cy Young Award winner, stifled the Diamondbacks on four hits and three walks while striking out six.
The left-handed ace won for the 13th time in 14 home starts for the National League West-leading Dodgers, who notched their fifth straight triumph.
Josh Lindblom tossed a scoreless eighth and Kenley Jansen picked up his fourth save as Los Angeles took the first of a two-game series against the D-Backs.
Arizona starter Ian Kennedy (3-3) was saddled with the loss after yielding three runs (two earned) on six hits over six frames.
Aaron Hill homered for the skidding Diamondbacks, who lost for the eighth time in nine games.