Kershaw (1-3) walked one batter and struck out five, and 57 of his 86 pitches were strikes. Three double plays helped his cause.
Jonathan Broxton recorded his fourth save.
Matt Kemp had two hits and his third-inning infield single extended his hitting streak to 14 straight games. Juan Pierre added two hits in the Dodgers' fifth win in their last seven games.
Nomar Garciaparra left in the sixth with left knee irritation.
Wil Nieves had two of the five hits for the Nationals, losers of six in a row.
Losing pitcher Jason Bergmann (1-8) allowed two earned runs and six hits in six innings.
Notable deaths of 2014 [PHOTOS]