Clevenger did not appeal and served his suspension Saturday.
"League Baseball put their hand down and I accept full responsibility," Clevenger said.
There were two bench-clearing incidents during Thursday's game, with Clevenger getting ejected after the second one. The rookie catcher shoved the Nationals' Michael Morse as players got heated and looked as if he tried to take a swing at Ian Desmond.
Trouble first started in the fifth inning when Cubs bench coach Jamie Quirk began shouting at Nationals third-base coach Bo Porter after Washington stole two bases and Jayson Werth took a swing at a 3-0 pitch. The Nationals were up five runs at the time and had been crushing the Cubs for three consecutive days.
An inning after the Quirk-Porter incident, which earned Quirk an ejection, trouble started again when pitcher Lendy Castillo nearly hit the Nationals' Bryce Harper with a pitch. Clevenger and Cubs pitcher Manny Corpas were ejected. The Nationals' Mike Gonzalez also was ejected.
"What happened happened," Clevenger said. "I have a responsibility to protect my players and my pitchers and I feel like I handled myself fine, but Major League Baseball put their hand down and gave me a suspension."
No other discipline is expected to be handed down for the incidents.