Thierry Henry did his best to open the scoring for the visitors in the first half, but it remained goalless at half time.
However, it did not stay that way for long after the restart, as Nathan Sturgis swooped from close range to put the hosts ahead after 47 minutes.
Oscar Pareja's men did not take their foot off the pedal and continued to put pressure on the Red Bulls, which eventually told in the 68th minute when Deshorn Brown rifled home from close range.
That took the sting out of the game and the Rapids went on to secure all three points, which sees them go fifth in the MLS Western Conference.
Meanwhile, the Red Bulls find themselves bottom of the Eastern Conference.