Lalas Abubakar and Braian Galvan also scored goals for the Rapids (9-4-4, 31 points), who are 6-2-3 over their past 11 matches.
Fafa Picault scored on a penalty kick for the Dynamo, who saw their winless stretch reach 12 matches (0-5-7).
Houston (3-7-9, 18 points) hasn't won since posting a 2-1 triumph over the Vancouver Whitecaps on May 22.
Colorado goalkeeper William Yarbrough registered a season-best nine saves as Houston held a 25-15 edge in shots. The Dynamo placed 10 on target to the Rapids' six.
Houston's Marko Maric made three saves as the Dynamo lost at home (3-1-4) for the first time this season.
Barrios' goal in the 60th minute gave Colorado a 2-1 lead.
Mark-Anthony Kaye threaded a cross-turf pass to the right to Barrios, who smacked the close-range right-footer past Maric. It was Barrios' fifth goal of the season.
Eight minutes later, Houston's Matias Vera had a chance to tie it but his right-footed blast was stopped by Yarbrough.
Galvan put the game away in the 80th minute as he received a pass from Cole Bassett and hammered a left-footed shot into the left corner of the net. It was Galvan's second goal of the season.
The Rapids struck five minutes into the match for the quick lead.
Jack Price took a corner kick and sent a cross into the box. Abubakar leaped and sent the header into the net for his second goal of the season.
Colorado had a chance to make it a two-goal lead nine minutes later but Barrios sent the header over the net.
Griffin Dorsey of the Dynamo had an opportunity to knot the score in the 26th minute but his close-range right-footed shot was saved by Yarbrough.
Houston landed the equalizer in the sixth minute of first-half stoppage time. Barrios was called for a foul in the box one minute earlier and Picault lined up for the penalty kick and easily booted a right-footer into the lower left corner for his fourth goal of the campaign.
Bassett twice hit the left goal post -- on a header in the 55th minute and a left-footed shot in the 74th.