Goals from Uruguayan import Diego Fagundez and substitute Charlie Davies cancelled out Sanna Nyassi's opener as the Revolution won 2-1 in front of 12,204 spectators at Gillette Stadium.
After extending their winning streak to four games, the in-form Revolution remain third in the East with 39 points from 27 matches.
New England are only three points adrift of second-placed Sporting Kansas City, who have lost four on the bounce.
The Fire are second from bottom in the East - seven points behind Philadelphia Union in the race for the final play-off spot.
And the Chicago franchise made a bright start in New England as they attempted to make it three games without defeat.
After heading an earlier effort wide of the post, Gambian forward Nyassi gave the visitors the lead in the 28th with a powerful header from Quincy Amarikwa's cross.
Nyassi had a golden opportunity to double the Fire's advantage after dispossessing United States international Jermaine Jones in the 30th minute but his effort hit the post.
Chicago were left to rue that chance as Fagundez restored parity five minutes before half-time, heading home from close range.
New England completed their comeback on the hour-mark, when Davies sprung the offside trap and placed the ball beyond Chicago goalkeeper Sean Johnson, having only been on the pitch for two minutes.
Columbus Crew are three points behind the Revolution after recording a comprehensive 3-0 win over lowly Chivas USA.
Justin Meram scored twice inside 15 minutes of the second half starting, while Bernardo Anor added a third for the home side with 10 minutes remaining at Columbus Crew Stadium.
Meanwhile, substitute Kalif Alhassan equalised four minutes from time as the Portland Timbers cancelled out a two-goal deficit to draw 3-3 at home to the San Jose Earthquakes.
USA international striker Chris Wondolowski and Cordell Cato helped the Earthquakes to a two-goal lead before Alvas Powell and former West Brom defender Liam Ridgewell hit back for Portland in the second half.
Wondolowski completed his brace and restored the visitors' lead with five minutes remaining, only for Alhassan to net the leveller a minute later.
The draw was enough for Portland to remain in the fifth and final play-off spot in the Western Conference on 35 points.
But the stalemate was a blow for the out-of-form Earthquakes, who are eight points adrift of the play-off places but have two games in hand.
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