Van Slyke hit a solo home run off Wade Miley (7-10) before leaving the game with an ankle injury in the fourth inning. Matt Kemp had a two-run double.
Kenley Janzen pitched a perfect ninth for his 38th save in 42 tries.
The Dodgers, sweeping the two-game series, climbed to a season-high 18 games above .500.
Kershaw reached double digit strikeouts for the third consecutive outing and his majors-best ERA dropped to 1.73.
"It is kind of hard to describe unless you are up there," Arizona's Mark Trumbo said, "but his heater plays as hard as anyone you will see, even though velocity-wise it isn't as hard. It is everything. It is his curveball when he gets on top of it. It starts way up there and goes way down, but everything is set up by his fastball."
And Kershaw was nowhere near his best, at least early on.
"It was a bad start, really," he said. "I felt guys were on base all night. You know, working in and out of jams, which is never fun to do. But I was fortunate to get some outs when I needed them. After about the fourth or fifth I settled in a little bit and was able to make it to eight."
That bases-loaded situation came with one out in the third, but Aaron Hill popped out and Trumbo struck out.
The Diamondbacks' Ender Inciarte tripled to lead off the fifth, then Kershaw retired the next 11 before Trumbo's two-out single in the eighth. Kershaw ended his night by striking out Alfredo Marte to end the eighth.
Los Angeles scored all three runs in the third.
Kemp doubled in two runs but was out by a considerable margin trying to stretch it to a triple. On Miley's next pitch, Van Slyke homered just over the left-field wall. Van Slyke is 8 for 18 with five home runs against Miley this season.
Arizona's Marte doubled down the left-field line to lead off the fourth. Jordan Pacheco followed with a single to left. Van Slyke fielded the ball and launched what should have been a routine throw back to the infield.
But he rolled his ankle as he threw, and the ball fluttered away, allowing the unearned run to score.
Mattingly said X-rays were negative and he didn't expect Van Slyke to be out more than a few days.
Hill and Trumbo, the No. 3 and No. 4 hitters in the Arizona lineup, each stranded five runners on base, four of them in scoring position. They were a combined 1-for-8.
"It stinks," Trumbo said.
Van Slyke said the ground gave way beneath his ankle as he planted to throw.