Chicago (7-8-15) won on the road for only the second time this year and improved its chances of earning a playoff berth. With 36 points, the Fire find themselves three points behind New York for the fourth and final wild-card spot.
Real Salt Lake played a man short from the 11th minute after Kyle Beckerman was sent off for head-butting.
Real Salt Lake (15-9-6) had already clinched a playoff berth.
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