Beckham left the Galaxy to join England for World Cup qualifying.
The Galaxy are two points behind the Rapids in the Western Conference and four points out of a playoff spot with two games remaining.
Edson Buddle opened the scoring for Los Angeles in the 36th minute. Colorado pulled even on Cory Gibbs' goal in the 43rd.
Colorado's Ugo Ihemelu inadvertently knocked a ball into his own goal on a Galaxy free kick in the 70th minute.
Brandon McDonald scored in the 80th minute to give the Galaxy a 3-1 lead before Colorado's Tom McManus scored on a penalty kick in the 88th minute.