Jaime Castrillon had given the Rapids the lead after just 10 minutes, finishing from close range.
Dallas took the ascendancy after going behind and managed to turn around the deficit before half-time.
Dallas continued to dominate and doubled their advantage just five minutes into the second half through David Ferreira.
The Rapids quickly responded with their own set piece, with Jeff Larentowicz driving home from distance.
Any hopes of a Colorado comeback were dented when they had scorer Castrillon sent off for kicking out at an opponent in an off-the-ball incident.
Dallas survived a series of late scares to earn the three points.