Hours earlier, pinch-hitter Matt den Dekker lined a two-run homer in the eighth inning that lifted the Nationals over the Dodgers 5-3 in the completion of a game suspended a day earlier after the fifth because of a lighting malfunction.
Kershaw (7-6), coming off a shutout of the Phillies, scattered three singles and didn't walk a batter. The lefty has won six straight starts against Washington, fashioning an 0.96 ERA in the process.
Bryce Harper struck three times against Kershaw, but hit a two-run homer in the ninth off reliever Kenley Jansen. It was Harper's 27th of the season.
Doug Fister (3-5) allowed four runs on nine hits over five innings.
Washington second baseman Danny Espinosa left the game after diving for Adrian Gonzalez's single in the ninth.
Gonzalez and Justin Turner each got three hits in the win, and Yasiel Puig drove in two runs for the Dodgers.
Gonzalez hit two home runs in the game that began Friday night and ended Saturday afternoon.
The Dodgers gave Kershaw a 1-0 lead on three consecutive singles in the second, the last coming on Puig's infield hit that deflected off Fister.
In the fifth, Turner and Puig sandwiched RBI singles around Andre Ethier's sacrifice fly to make it 4-0.
The series opener was delayed three times Friday night by a bank of lights on the third base side that kept going out, and finally stopped with Washington ahead 3-2. The Nationals said Saturday that a faulty circuit breaker caused the outage.
When the suspended game resumed, Gonzalez hit his second homer of the game, a tying shot in the sixth.
Den Dekker hit his second homer of the season and the first of his career as a pinch hitter. The two-out drive into the second deck came against Pedro Baez (2-2).
Yunel Escobar homered and doubled twice for Washington.
Casey Janssen (1-2) pitched one inning, and Drew Storen picked up his 28th save.
Nationals: INF Anthony Rendon (left quadriceps strain) was scheduled to begin a rehabilitation assignment Saturday night for Class A Potomac, according to manager Matt Williams. . OF Jayson Werth (broken left wrist) was slated to play the outfield in his third game with Potomac. . 1B Ryan Zimmerman (left foot plantar fasciitis) took batting and infield practice prior to the game.
Dodgers: RHP Zack Greinke (8-2, 1.39 ERA), the major league ERA leader, pitches the series finale in a matchup of NL All-Stars. Greinke is 3-1 with a 2.22 ERA in five career starts versus Washington.
Nationals: RHP Max Scherzer (10-7, 2.11 ERA) is 2-3 with a 3.51 ERA in nine games, including seven starts, against the Dodgers.