Jay Heaps' side stepped up with a big win when it mattered, thrice taking the lead before ending Columbus' playoff hopes.
Columbus needed a win to mathematically remain in playoff contention, while a loss or draw would make it exceedingly difficult for New England to claim a postseason berth.
A.J. Soares scored the only goal of the first half in the 32nd minute when his flicked header redirected Kelyn Rowe's free kick into the net.
Columbus leveled it just before the hour mark. Aaron Schoenfeld, at the top of the box with his back to goal, laid off a pass to Dominic Oduro, charging down the right channel. Oduro used his pace to get past the Revs back line, and roofed a shot over Matt Reis at the near post.
New England was awarded a penalty in the 69th minute. Tony Tchani hauled down Andy Dorman in the box, and Chris Tierney dispatched the spot kick with aplomb to restore the Revolution's lead.
The advantage lasted barely a minute. Charging down the other end, this time it was Oduro's turn to feed Schoenfeld at the top of the box, and the striker turned and picked out the far top corner to equalize again.
But with their season on the line, the Revs refused to be pegged back. Dimitry Imbongo picked out Dorman on the left wing, and Dorman's cross found Diego Fagundez, who finished to put New England up 3-2 and set a record for MLS goals scored by a teenager.
With the victory, New England (13-11-9) jumps to 48 points and fifth place. But these teams meet again next weekend, this time in Columbus, and the Crew (12-16-5) will have the opportunity to play spoiler to the team that ended its season.