The Revolution seemed about to pick up three points on the road until Jaime Moreno took a free kick from outside the penalty area. The ball sailed to Olsen at the far post and he put a header through traffic that just slipped by the post and into the net.
It was the 29th MLS career goal for Olsen, who is in his 11th season with United.
United outshot New England 20-4, but fell behind when Shalrie Joseph scored early in the second half.
New England is tied for the top of the Eastern Conference with Chicago at 2
2. United is 1-1-3.