Kershaw didn't allow a baserunner until Dylan Carlson's one-out single in the fifth. The three-time NL Cy Young Award winner gave up two hits, struck out nine and walked none, throwing 88 pitches.
“Four innings, you don't really think about (being perfect) but it's nice to get off to a good start like that and kind of get into that rhythm,” Kershaw said.
Back for his 16th season on a one-year deal, the 35-year-old Kershaw (5-1) won his fourth consecutive start. He has yielded one earned run over his last three outings, a span of 20 innings. His ERA dropped to 1.89.
“They were swinging a lot, which kept my pitch count down, which was good,” Kershaw said. “I mean it's a tough lineup. (Paul) Goldschmidt and (Nolan) Arenado are two of the toughest at-bats I have in the game.”
Austin Barnes hit an RBI single in the second inning for Los Angeles, which will try for its first three-game series sweep Sunday. Evan Phillips worked around a walk and an error by shortstop Chris Taylor in the eighth, and Brusdar Graterol finished the three-hitter for his second save.
Cardinals left-hander Jordan Montgomery (2-4) had his second consecutive hard-luck defeat after taking the loss Monday at San Francisco when he gave up one unearned run. He struck out seven while pitching 6 2/3 innings of five-hit ball, but he lost his fourth consecutive decision.
“Monty was good, man,” Cardinals manager Oliver Marmol said. “He went toe to toe with (Kershaw). The sinker was in play, seven punchouts overall. He was determined to have a good game. We saw a really good version of him tonight.”
The Cardinals have lost five of six. They are 2-7 on a 10-game West Coast trip that started in Seattle last weekend.
Barnes drove in James Outman with his two-out single to center in the second.
“It felt good and obviously I have been struggling a little bit,” said Barnes, who is batting just .093. “Just to come through, get a run on the board, especially when (Kershaw) is on the mound. When we're downhill and have a lead he's really, really good.”
After Carlson's single, Kershaw retired five straight batters before Goldschmidt singled to lead off the seventh.
“I think I've said it before, maybe even this year, but tonight was his best night for me (this season) as far as stuff, start to finish,” Dodgers manager Dave Roberts said of Kershaw. “I thought the fastball (was good), the curveball, the lateness of it was as good as I've seen it. The slider. Just the way he tunneled tonight was exceptional.”
Carlson contributed on defense with a running, over-the-shoulder catch of Mookie Betts' drive to the center-field warning track in the fifth.
Arenado, who entered batting .248 with a .625 OPS, was moved to the No. 3 spot in the St. Louis order for the first time this season after batting cleanup for all but one game. He went 0 for 4.
The Dodgers stole three bases, including two by Freddie Freeman in the first multi-steal game of his career.
Cardinals: After he was a late scratch Friday as a precaution following mid-back tightness in batting practice, left fielder Tyler O’Neill was back in the lineup. ... RHP Adam Wainwright (groin) is expected to make his third rehab start Sunday for Triple-A Memphis.
Dodgers: SS Miguel Rojas, who has not played since April 18 because of a hamstring injury, began a rehab assignment with Single-A Rancho Cucamonga on Saturday. He could return to the Dodgers by Monday.
Cardinals: RHP Jake Woodford (1-2, 5.47 ERA) starts on Sunday. He has allowed seven home runs over his five starts this season.
Dodgers: RHP Noah Syndergaard (0-3, 6.58 ERA) will make his sixth start for his new club while still looking for a win.
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