Movsisyan, who entered the game in the 82nd minute, got his fifth goal of the season and put the Wizards into second place in the MSL Eastern Conference, one point behind New England.
Chivas USA scored first in the 17th minute with a Jonathan Bornstein goal. Kansas City tied it before halftime as Kerry Zavagnin scored.
Kansas City took the lead after halftime through Davy Arnaud, but John Cunliffe's first-ever MLS goal tied the score again at 2-2.
Notable deaths of 2014 [PHOTOS]