Arizona had opportunities early, with that bases loaded with one out situation in the third and a leadoff triple by Ender Inciarte in the fifth. But Aaron Hill and Trumbo -- the No. 3 and No. 4 hitters -- stranded all of them.
"We actually had better opportunities with runners in scoring position than the Dodgers did tonight," Diamondbacks manager Kirk Gibson said, "but he shut us down."
Kershaw (16-3) allowed six hits, retiring 12 of the final 13 batters he faced, seven by strikeout, to improve to 13-1 in his last 16 starts.
The left-handed ace struck out 10 and walked two.
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 straight outing and his major league best ERA dropped to 1.73.
That bases-loaded situation came with one out in the third, but Hill popped out and Trumbo struck out.
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.
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.
Van Slyke said the ground gave way beneath his ankle as he planted to throw.
"They've got bad grass here," he said.
Television replays showed that Van Slyke's ankle bent to almost a 45-degree ankle, but he was able to walk off the field.
"That side of my ankle's loose, so I think it will be all right,' he said. "I think it just went further than It normally does."
A LOT BETTER THIS TIME
Kershaw had his only truly bad outing of the season in his last appearance in Arizona. On May 5, he lasted just 1 2/3 innings, allowing a season-worst seven runs in the Diamondbacks' 18-7 rout.
Dodgers: 1B Adrian Gonzalez had the night off with a stiff back. ... LHP Hyun-Jin Ryu (hip strain) pitched a simulated game Wednesday and is expected to rejoin the rotation Sunday or Monday.
Diamondbacks: RHP Daniel Hudson, recovering from two Tommy John surgeries, pitched a perfect inning at Triple-A Reno on Wednesday night.
Dodgers: Los Angeles has Thursday off before opening a three-game series at San Diego. Dan Haren (11-10, 4.44 ERA) goes for the Dodgers against Andrew Cashner (2-6, 2.43).
Diamondbacks: Arizona is off Thursday before beginning a three-game series against Colorado in a matchup of the teams with the two worst records in the National League.