Ecuadorian forward Joao Plata struck the winner for Real in the third minute of second-half stoppage time, volleying Alvaro Saborio's cross into the net from close range to keep the Utah-based club within touching distance of Western Conference leaders Seattle Sounders.
Chicago had led 2-0 at half-time but fell apart in the closing stages to extend their winless streak to four games.
The Fire took the lead in the 22nd minute in Chicago through Mike Magee, who tapped home the rebound after Real's Nick Rimando could only parry Harrison Shipp's low driven effort.
On the half-hour mark, Juan Luis Anangono doubled the Fire's advantage with a diving header, converting Greg Cochrane's cross from the left.
With Chicago pushed back, Real attacked down the left and Olmes Garcia's cross somehow found its way to Saborio at the back post, who controlled the ball with his thigh and then thumped it into the net.
With five minutes remaining in the first half, Seattle were given a penalty when Amobi Okugo handled the ball in the box but Osvaldo Alonso's spot kick was parried by Zac MacMath onto the post and out.
But the West's leaders managed to recover and clinch three points with Obafemi Martins equalising in the 61st minute from close range, and Chad Marshall headed home the winner from a corner with six minutes left.
