Harris took a pass from Edson Buddle and made sure with a left-footed shot from the middle of he box.
One of the saves made by Irwin came on a penalty shot in the 8th minute taken by Alvaro Saborio.
It was Colorado's first victory of the season following three losses and two draws. The Rapids began the night in next-to-last place in the Western Conference.
Real Salt Lake fell to 2-3-1 in the young season.
The goal for Harris was his first of the year and only the fifth Colorado has managed for the entire campaign.