Jhonatan Solano's solo shot in the fourth broke a 3-3 tie and Zimmerman's blast in the next frame off Madison Bumgarner (10-5) put the Nationals ahead by three.
Morse, batting next, made it 7-3 with a long, bases-empty clout and National League East-leading Washington went on to clinch its three-game set at Nationals Park.
Edwin Jackson (5-4) got the victory despite allowing four runs on five hits through 5 2/3 innings.
Bumgarner ended up yielding seven runs on nine hits over five full frames for the Giants, who came into the contest tied with the Los Angeles Dodgers for the NL West lead.
Pablo Sandoval went 3-for-4 with a two-run homer and RBI double in the loss.