Kansas City has not allowed a goal in 429 consecutive minutes, last surrendering one to Toronto on March 9.
The winning goal came on a cross from Soony Saad and Bieler converted it with a left-footed blast.
Kansas City moved into second place in the Eastern Conference with 11 points. D.C. United remained in eighth place with just four points in five matches.
Police: Sword-wielding man demanded free tacos
Sign language interpreter at Mandela service called out as fake on Twitter