The game was scoreless until the 56th minute, after which the team squads combined to score four goals in a 10-minute span.
Colorado's Atiba Harris scored in the 56th minute, and Perez responded with the first of his two goals 5 minutes later to forge a 1-1 draw.
Harris struck again for a Rapids in the 64th minute after finishing a cross from Chris Klute.
But Perez got the final say for Western Conference-leading Dallas (8-2-4), firing home a low drive from 20 yards out.
The Rapids (5-4-5) drew even with Los Angeles for fourth place in the West with 20 points.