Both benches emptied in the sixth inning following a series of incidents that began in the fifth, during which both Danny Espinosa and Zimmerman stole second base with the Nationals holding a 7-2 lead.
Jayson Werth, batting with the bases loaded and two out in the fifth, also swung at a 3-0 pitch and the accumulation of what Chicago bench coach Jamie Quirk felt was bad baseball etiquette caused him to start yelling at Washington third base coach Bo Porter.
Quirk became so upset he was kicked out of the game by home plate umpire Jerry Layne.
In the sixth inning, Chicago pitcher Lendy Castillo threw behind Washington's Bryce Harper and players from both teams spilled onto the field.
No punches were thrown, but pushing and shoving took place before order was restored.
Suzuki hit a three-run homer in the second. Zimmerman drove in a run with a ground out in the first and two more in the fourth on a single.
The win went to Jordan Zimmerman (10-8), who allowed two runs on five hits over seven innings with nine strikeouts. Justin Germano (2-6) took the loss after giving up six runs on seven hits in four frames.
Washington won its fifth in a row and moved 7 1/2 games in front of Atlanta at the top of the National League East.