In the semi-finals of the joint Liga MX-MLS tournament reigning MLS champions Columbus Crew will face the Philadelphia Union while the Rapids will face Los Angeles FC.
America, the most decorated team in Mexico, were playing a 'home' game in Carson, California, home of the Los Angeles Galaxy and attracted thousands of Mexican-American supporters to back them.
But they were left disappointed after their goalless draw, with few chances for either team, went to a shoot-out which the Rapids won 9-8.
The marathon shoot-out ended up in sudden death with both teams' goalkeepers taking kicks and after Colorado's Zack Steffen slotted home, Luis Malagon scuffed his effort wide.
The other Mexican club in the last eight, Mazatlan, also exited via a shoot-out, losing at the Philadelphia Union. Philadelphia had found themselves a man down just before half-time when forward Tai Baribo was harshly sent off for a second yellow card offence.
But two minutes later, Danish forward Mikael Uhre put the Union ahead when he broke down the left and cut inside before finishing with a deft chip over Mazatlan keeper Ricardo Gutierrez. The Liga MX team drew level in the 59th minute when Alonso Escoboza rose to head in a cross from Ramiro Arciga.
But despite their numerical and possession advantage, Mazatlan were unable to find a winner with Philly keeper Andre Blake ensuring the game went to a shoot-out after a brilliant reflex save kept out a goalbound header from Facundo Almada.
Jamaican Blake was the hero in the shoot-out with two superb saves as Jim Curtin's side won 4-3 to make it to the last four for successive seasons.
Columbus marched into the last four with a 4-3 win on penalties over New York City after the game ended 1-1 while a ten-man Union also needed a shoot-out to beat Mexican team Mazatlan.
Just 14 seconds into the game in Columbus, visitors New York took the lead when Alonso Martinez robbed Crew defender Rudy Camacho and burst into the box before slotting home.