The Rapids extended their unbeaten streak to four games (3-0-1) on the heels of a three-game losing skid.
Pressure on the New England goal in the 66th minute led to the mistake by Goncalves, who attempted to send the ball over the end line and concede a corner kick but instead drove it just inside the right post for a Colorado goal.
New England thus wasted a 1-0 lead it obtained when Juan Carlos Toja scored in the 15th minute. Colorado tied it in the 62nd minute on a long, left-footed shot from Nick LaBrocca.
The Rapids' unbeaten streak has boosted them to 8-7-6 for the season and into a tie for fifth place with Los Angeles in the Western Conference.
New England has lost its last two games and has only one win in its last six, leaving the Revolution in sixth place in the East.