Zimmermann (7-1) racked up his fourth victory in four starts, working around nine hits to allow just two runs. The right-hander didn't issue a walk and struck out five.
Zimmerman plated Denard Span with a first-inning fielder's choice, opening the scoring, and later sent home Zimmermann and Steve Lombardozzi with a two-run double in the third.
Adam Laroche also knocked in a pair of runs for Washington, which opened a 10-game California road trip with a victory.
Dodgers starter Josh Beckett (0-5) remained winless in eight starts to begin the season, allowing four runs (two earned) on three hits before exiting with a groin injury after three innings.