Melia made game-saving stops at the end of the first half and 30 minutes into the second, and finished with three total saves while posting his third shutout of the season and 55th of his MLS career.
The draw allowed Sporting Kansas City to move into first place in the Western Conference, one point ahead of the Seattle Sounders.
Colorado goalkeeper William Yarbrough recorded his eighth shutout of the season -- tied for the MLS lead -- without making a save. The Rapids are unbeaten in their last eight home matches (5-0-3).
Sporting Kansas City failed to put any of their six shots on target. Colorado took 13 shots with three on target.
Melia's first big save came in the first minute of first-half stoppage time. Jack Price had operating room for Colorado but his close-range right-footed shot was thwarted by Melia, who made a kick save with his left foot.
The second superb stop occurred in the 75th minute. Price took the corner kick and sent a cross toward the goal for Diego Rubio, who sent a short-range header toward the net, but Melia got his hand on the ball to tip it over the net.
The Rapids had two other strong chances in the second half.
In the 73rd minute, Cole Bassett spotted Jonathan Lewis open to the left of the net and got him the ball. Lewis sent a soft left-footed ground shot that got past Melia but it trickled just wide of the right corner of the net.
In the first minute of second-half stoppage time, Rubio sent a header just over the top of the net.
Sporting Kansas City's best chance of the match came in the 51st minute but Grayson Barber's right-footed blast sailed over the net.
Alan Pulido had an opportunity 15 minutes later for SKC but his right-footed shot was wide to the right.
Colorado's Michael Barrios had a good opportunity for the Rapids in the 28th minute but he lifted his short right-footed shot high and it landed on top of the net.