With qualification to the next round in sight, the Galaxy have now lost two CCL matches in a row, the latest coming against Alajuelense.
Pablo Antonio scored the only goal of the game with a stunning left-footed free kick from 27 yards midway through the first half.
The Costa Rican side put the ball past Josh Saunders again in the second half, but the play was correctly ruled offside.
For the Galaxy, Landon Donovan, David Beckham - making his first ever appearance in Central America - and Robbie Keane completed all 90 minutes. Los Angeles will miss Beckham in the rematch against Morelia next week after the Englishman collected a yellow, his second of the group stages, in the 52nd minute against Alajuelense.