"It was a fantastic header from Drew and it capped a wonderful performance for him," Colorado coach Gary Smith said. "And a lovely cross form Jamie Smith."
Moor's first goal of the year ended a run of seven straight games at home without a win against the Crew for Colorado (6-3-1). The Rapids last beat the Crew in Denver on April 6, 2006.
"We'd lost two of those games, 1-0, while I've been here," Smith said. "It's nice to have it come our way once in a while."
The Rapids won their third straight overall, all by 1-0 scores. Goalkeeper Matt Pickens stopped three shots for his fourth shutout.
"A blue-collar win," said Moor, whose goal was his first since last June 13. "Tonight was my night to score."
William Hesmer had four saves for Columbus (6-2-3), which is 0-2-1 in its last three games.
Crew coach Robert Warzycha remained at home to attend his daughter's high school graduation.
"The game can be cruel at times," acting coach Mike Lapper said. "I thought that was a 0-0 draw game. I was pleased with everybody's work ethic, attitude and effort.
"It came down to individual duels and theirs was more effective than ours."
The game appeared headed for a scoreless draw until defender Jamie Smith gained control on the Columbus side of midfield. He sent a pass toward the left side of the crease and Moor sent a header into the left side of the net.
Hesmer had to catch a misfired pass in the game's opening minute, but his defense kept his work to a minimum, blocking six shots and denying numerous passes towards the crease in the game's first 35 minutes.
Meanwhile Columbus managed several scoring chances in the same time frame. Steve Lenhart got a shot off from the slot in the 11th minute, but Rapids defender Julien Baudet batted it away before it reached Pickens.
Eddie Gaven's shot from a severe angle was grabbed by Pickens in the 24th minute. Robbie Rogers sent a free kick from 30 yards out well past the right goal post.
Omar Cummings had the first shot for Rapids in the 36th minute, but it was caught by Hesmer.
Referee Kevin Stott issued two yellow cards in the first 45 minutes. Columbus midfielder Adam Moffat drew one after upending Conor Casey in the 40th minute and Casey received one for arguing a non-call when Cummings was tripped up just before the end of the first half.
Two more shots by the Rapids were blocked to start the second half before Hesmer stopped Pablo Mastroeni's low drive from outside the scoring box in the 52nd minute. Colin Clark missed everything on a free kick in the 56th minute for Colorado at the start of the second half.
Minutes later Columbus' Guillermos Barros Schelotto had a free kick stopped by Colorado's wall. Lenhart set up Gaven for a shot from just inside the scoring box but Pickens caught the shot in the 64th minute.
The teams missed chances seconds apart. Schelotto kicked a shot over the crossbar in the 74th minute and Crew defender Chad Marshall denied Cummings' attempt to sneak the ball inside the right post.