All of the scoring came in the second half after the teams combined for just two shots in a listless opening 45 minutes.
An outstanding play by defender Paul Broome gave Dallas a 1-0 lead in the 53rd minute.
Pushing forward from his position on the left side of the backline, he intercepted a distribution pass from goalkeeper David Kramer and darted into the left side of the penalty area. With Colorado's attention focused on his advance, Broome slotted the ball to unmarked forward Ronald Cerritos, who easily tucked the ball home.
The Rapids tied it in the 62nd minute. A quick combination between Carlos Valderrama and Chris Henderson left first-half substitute Chris Carrieri with a tap-in from six yards.
Positioned at the edge of the penalty box, Valderrama nodded a short clearing header from defender Tenywa Bonseu back into the box, from where Henderson popped the ball over Steve Morrow to set the table for Carrieri's easy finish.
Continuing to pressure the Burn defense, Colorado got the go-ahead goal in the 76th minute on a penalty kick by Mark Chung. The veteran midfielder was hauled down by Chad Deering deep in the 18-yard box and rifled the ensuing penalty kick high into the back of the net.
As referee Brian Hall prepared to blow the final whistle in stoppage time, Dallas netted the stunning equalizer. A curling ball into the box by the Antonio Martinez was desperately cleared by a sliding Jeff Stewart. The rookie's effort backfired, however, when it fell right to the foot of Morrow, who drilled a 20-yard shot past David Kramer for the equalizer.
Dallas narrowly missed two chances in the extra session as a brilliant individual effort by Antonio Martinez was stuffed by Kramer in the 94th minute and second-half substitute Edward Johnson blasted a shot off the crossbar in the opening moments of the second overtime.