Kansas City came into the game facing a one-goal deficit in the two-match series and managed to finish the regulation 90 minutes of Wednesday's game with a 2-1 advantage.
With the aggregate goal total deadlocked, a 30-minute overtime was played and Kansas City emerged with the victory when Bieler scored from in close during the 113th minute.
Kansas City will take on Houston in the conference championship series, the first leg of which will be played in Houston on Saturday. Houston stunned New York earlier Wednesday with the second game of that series also requiring overtime.
Needing to win Wednesday's game by two goals to advance, Kansas City came out aggressive and scored in the 41st minute on a goal by Aurelien Colin. New England tied it in the 70th minute when Dimitry Imbongo Boele scored and if the Revolution could have prevented another Kansas City goal they would have advanced.
Seth Sinovic's goal in the 79th minute kept Kansas City's hopes alive and Bieler finished things off with his overtime score.
The Western Conference semifinals will be completed on Thursday. Portland will take a 2-1 advantage over Seattle into their second leg and Los Angeles will hold a 1-0 lead over Real Salt Lake.