Kennedy allowed just three hits and three walks while striking out six over 5 1/3 innings of work in his finest start of the spring.
Kershaw, meanwhile, allowed a pair of runs on eight hits while striking out five in five-plus innings.
2010 Spring Training - null
Sights & Sounds
Spring Training Info
Kershaw had been scheduled to start the Dodgers' afternoon game on Monday, but was pushed to the night game when the day game was canceled due to rain.
The D-backs got on the board in the fourth after Chris Young doubled with one out and scored when Miguel Montero followed with a single to left.
Montero scored three batters later on Xavier Nady's single to center, as the D-backs grabbed a 2-0 advantage. David Winfree drove in Juan Miranda with a single in the seventh to complete Arizona's scoring.
The Dodgers threatened in the second, as they loaded the bases with one out. Kennedy, though, got Kershaw to strike out swinging before Dee Gordon grounded out to end the frame.
After walking Ivan De Jesus to lead off the third, Kennedy retired 10 straight before walking Matt Kemp with one out in the sixth. He was then replaced by lefty Joe Paterson, who got James Loney to ground into an inning-ending double play.
Dodgers up next: In an attractive matchup of right-handers, Chad Billingsley starts for the Dodgers on Tuesday against the Cubs and Ryan Dempster at 1:05 p.m. PT. The lineup behind Billingsley is likely to be without Andre Ethier, Matt Kemp, James Loney and Jay Gibbons, as they all played Monday night. Jonathan Broxton and Matt Guerrier are expected to follow Billingsley. Tim Redding had his scheduled start for Monday's day game scratched, and he will instead throw a bullpen session.
D-backs up next: The D-backs play host to the Indians on Tuesday at Salt River Fields at Talking Stick at 1:10 p.m. MST. Aaron Heilman will get the start for the D-backs as the right-hander continues in his push to win one of the final two spots in the rotation. Heilman's last regular-season start was May 15, 2005, and he is the only pitcher in Major League Baseball to appear in at least 70 games for the last five seasons.