Finlay ran down a long ball from Justin Meram, shrugged off a defender, and hammered a shot into the back of the net.
Federico Higuain opened the scoring for the Crew (6-7-8) in the 43rd minute, bending in a free kick around the wall from 23 yards out. New England's Kelyn Rowe missed on a left-footed shot in the 50th minute, but his ensuing corner kick to the head of A.J. Soares tied it at 1.
New England (7-11-2) is one short of the club record for consecutive losses set in 1997-98. The Revolution have scored four goals during the slide.
Wil Trapp received a yellow card in the 41st minute and will miss the Crew's next match on an accumulation suspension.