Austin (10-4-4, 34 points) improved to 3-0-1 in its past four matches, holding on for the win despite playing the last 15 minutes with 10 men after Kipp Keller was sent off for receiving a second yellow card.
Colorado (5-8-4, 19 points) lost its second straight and has dropped four of its past six games.
Lalas Abubakar put the Rapids ahead 1-0 in the 19th minute off a free kick. Bryan Acosta directed the kick into the box, and Abubakar stretched to get his foot under the ball at the top of the goal area.
It was Colorado's fifth shot attempt and second on goal, while Austin had zero attempts to that point.
But the visitors soon found life.
Ethan Finlay got behind the Colorado defenders at the top of the box and lifted a shot past goalkeeper William Yarbrough, who came far out of the goal area, to cut the deficit to 2-1 in the 28th minute.
The Rapids came close to regaining their two-goal lead late in the half. Lewis nearly had his second in the 39th minute, but his shot deflected just past the right post. A minute later, Lucas Esteves directed a long shot that curled just wide, and in the 45th minute, Esteves' header just missed the net.
Austin's Sebastian Druissi leveled the score moments before halftime. Jon Gallagher lifted a feed to Druissi that landed right at the striker's foot at the top of the goal area, and Druissi tied it 2-2 in the first minute of first-half stoppage time.
Maximiliano Urruti appeared to score to give Austin a 3-2 lead in the 52nd minute, but the goal was overturned after video review due to a handball.
Urruti got it back in the 59th minute. Druissi dummied and Urruti finished with a strike from the center of the box to put the visitors ahead 3-2.
Abubakar nearly tied it in the 73rd minute with a header off a corner, but his attempt hit the post.