Sidelined since late March by a muscle strain in his upper back, Kershaw (2-0) scattered nine hits and fanned nine without a walk while displaying little rust from a layoff of more than six weeks.
The Dodgers provided the reigning NL Cy Young Award winner with plenty of support, with Hanley Ramirez going 3-for-5 with a homer and two RBI and Drew Butera belting a three-run shot in the rout. Dee Gordon added two hits and a pair of runs scored, while Andre Ethier finished 2-for-5 with an RBI.
Washington's Blake Treinen (0-1) allowed three runs in five-plus innings in his first major league start, though none were earned due to an error the rookie committed in the sixth.
Jayson Werth went 4-for-5 with an RBI for the Nationals, who posted a 4-0 triumph in Monday's opener of this three-game set. Anthony Rendon collected three hits in the loss.
Treinen matched zeros with Kershaw through the game's first five innings, but hurt his cause by mishandling the Dodgers ace's slow roller in front of the mound that eventually led to a three-run top of the sixth.
Los Angeles loaded the bases without getting a ball out of the infield, as Gordon beat out a chopper to first and Carl Crawford followed with a bunt single before Ramirez fought off a inside pitch for a base hit that broke the tie.
Craig Stammen came on in relief to strike out Matt Kemp, but the Nationals couldn't turn two on Ethier's run-scoring grounder and Juan Uribe made it 3-0 with a broken-bat single that brought home Crawford.
Another defensive lapse enabled the Dodgers to extend the margin in the seventh. After Gordon tripled into the gap in right center, Scott Hairston dropped Crawford's routine fly ball to left that would have been the third out.
Kershaw, meanwhile, was very sharp in his first outing since Los Angeles' season opener in Australia on March 22. The standout lefty threw 23-of-28 first-pitch strikes, didn't give up an extra-base hit or let a runner past second.
Ramirez mashed Ross Detwiler's first pitch of the evening over the wall in straightaway center in the eight for a 5-0 lead. Butera's homer occurred later in the inning, putting the game well out of reach with a blast to left with Kemp and Ethier aboard.
Washington finally got on the board when Rendon tripled in the bottom of the eighth and came around on a Werth single, with Jose Lobaton tacking on a two- run single off Chris Perez before the frame closed out.
Celebrity Breakups and divorces of 2014 [PHOTOS]