The Galaxy has won consecutive games for the first time this season and moved within eight points of the final playoff spot.
Kansas City's Jimmy Conrad was sent off in the 57th minute for making a vicious tackle on Peter Vagenas. Davy Arnaud also drew a red card in the final minutes, ending Kansas City's hopes for a comeback.
The Wizards, who hold down the eighth and last playoff position, have lost four of their last six. They are in front ninth-place Columbus by only five points.