With 19 minutes remaining, the visitors seemed set to leave Dick's Sporting Goods Park with a draw, but their hopes were dashed when Donovan Ricketts brought down Deshorn Brown in the area, a challenge for which he earned a red card.
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Vicente Sanchez converted the spot kick to give the Rapids a vital lead that they soon increased.
It came from a penalty moments later as Sanchez was brought down by substitute Andrew Weber and Brown was given the chance to seal the points from 12 yards out.
However, he saw his spot kick saved by Michael Harrington only for the forward to react first to the rebound and rifle home from close range.