The home side twice squandered the lead before 18-year-old midfielder Diego Fagundez sealed the points with just under 15 minutes remaining at Gillette Stadium.
As a result, Revolution are now in fourth place, level on 48 points with Houston Dynamo and two points ahead of sixth-placed Chicago Fire with one game left.
However, Frank Klopas's men are currently playing their penultimate game of the season against Toronto FC.
He could only keep the Crew at bay for so long, as Dominic Oduro lashed into the net to restore parity just before the hour mark.
Chris Tierney stepped up and made no mistake, but a smart turn and finish from Aaron Schoenfeld two minutes later saw the teams level once again.
The points were decided when Fagundez rifled in Dorman's supply to score his 17th goal in the MLS, which is a new record for a teenager.