After the first leg ended 1-0 in California, Colorado's Designated Player Shkelzen Gashi scored an absolute cracker from range in the 36th minute to tie up the overall scoreline at 1-1, which eventually forced a shootout.
Liverpool legend Steven Gerrard stepped up first for the Galaxy and beat former Everton goalkeeper Tim Howard from 12 yards before ex-Reading and Wolverhampton Wanderers striker Kevin Doyle drove his effort through Galaxy keeper Brian Rowe.
Former Barcelona, Tottenham Hotspur and Villarreal attacking midfielder Giovani dos Santos then fired his attempt over the bar for the Galaxy before Cole joined him in missing after Sebastien Le Toux and Marco Pappa converted emphatically for the home side.
Jeff Larentowicz needed to score to keep LA in it but Howard glanced his attempt into the upright as Colorado ended the play-off run of the five-time MLS Cup winners.
The match was Liverpool legend Gerrard's final one in a Galaxy shirt as he prepares to move on before the new year.