New England took the lead at the Columbus Crew Stadium after 17 minutes through Ryan Guy and doubled their advantage six minutes later when Jerry Bengtson found the target.
That sparked Crew into life and the home side replied strongly in the first half.
Federico Higuain pulled one back in the 26th minute from a free kick and Jairo Arrieta made it 2-2 six minutes later with a left-footed shot from the left side of the penalty box after being released by Milovan Mirosevic.
Crew went into the half-time interval 3-2 up after Higuain scored in the 43rd minute from yet another stunning free kick.
The hosts maintained that lead until the 80th minute when they conceded a penalty.
Julius James was adjudged to have pulled down Diego Fagundez inside the 18-yard box and Lee Nguyen successfully converted the spot kick.
However, Crew had the last laugh and bagged the win in this Eastern Conference clash with their fourth goal of the match in the 86th minute.
Tony Tchani played a looping through ball and Arrieta produced a low shot into the New England net to complete his brace.