The home side got off to the perfect start when they took the lead through Harris when he bundled the ball into the net after the visitors had failed to clear their lines from an Edson Buddle cross.
Real Salt Lake had the chance to get level straight away when they were awarded a penalty. Dillon Powers brought down Kyle Beckerman in the area but Clinton Irwin made a good save to deny Alvaro Saborio from the spot.
After the break, Irwin did well to deny Saborio again when he got down low to his right to stop the striker's powerful effort reaching the bottom corner after he turned his man on the edge of the area.
Brown went inches away from getting his second of the game in the 73rd minute when he got in front of his marker in the area and prodded Chris Klute's cross at goal, but the ball spun wide of the post.
The Rapids looked the side more likely to score again and Rimando had to do well to turn Jamie Smith's effort over the bar when he drove into the area and fired a powerful shot at goal.
Real Salt Lake tried late on to get an equaliser but the closest the game was through Saborio when his header went wide of the target, as the home side held out for the win.