Kansas City had lost its three previous games and had gone 340 minutes without a goal before Bunbury fired a shot between the legs of goalkeeper Will Hesmer.
Bunbury ran onto a pass from Ryan Smith at the top of the penalty area to produce the goal.
Kansas City's Jimmy Nielsen made seven saves for his sixth shutout of the season. It was the first Kansas City win over Columbus since September 2007.
The Wizards improved to 4-8-3 and Columbus fell to 8-3-4.