Federico Higuain missed a penalty kick for the Crew in the first half at Crew Stadium, and RSL went in front in the 56th minute through Luis Gil.
But Finlay came off the bench in the 77th minute and found the net 11 minutes later with a nice chip to salvage a point for the hosts.
The result leaves the Crew with just one win from their last 11 games, while Salt Lake has claimed only two points from their three most recent outings.
Columbus had a great chance to snatch the lead in the opening 15 minutes when RSL's Rich Balchan pulled down Crew midfielder Agustin Viana inside the penalty area.
However, Higuain stepped to the spot and hammered a shot well over the crossbar before leaving the game late in the first half with an injury.
The opening goal arrived in the 56th when a shot from Kenny Mansally was saved by Crew goalkeeper Steve Clark, who watched Gil poke home the rebound.
From there, Columbus was the more dangerous team and the hosts pressed hard for an equalizer, only to be turned away time and again by some great goalkeeping from Jeff Attinella.
The Crew kept pushing and finally the hosts were rewarded two minutes from time as Ben Speas picked out Finlay on the right wing, allowing him to get in on goal where he scored with a nice chip over the head of the onrushing Attinella.
Notable deaths of 2014 [PHOTOS]