Mapp, a second-half substitute, led a furious Union comeback after it seemed the Revolution had the game under control.
The Union (5-11-5) tied it in the 82nd minute when Jack McInerney volleyed home a perfect service from Mapp into the area.
And Mapp notched the 92nd-minute winner when he got the ball at the top of the area before one-timing his first goal of the season past New England goalkeeper Bobby Shuttleworth.
The Revolution (6-12-3) scored in the 31st minute when Ilija Stolica kicked home a loose ball after Marko Perovic's corner kick.
Swim Week Miami Beach 2014 [PHOTOS]