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.
Scarlett Johansson steps out with fiance after pregnancy reveal
Teacher apologizes for showing sexual image of herself in class