LA became the eighth and final team to book a CCL quarters slot thanks to its first away win in the competition this year.
Los Angeles, Morelia and Alajuelense all finished on 12 points atop Group A, but Costa Rican side Alajuelense was eliminated based on goal difference in the matches between the three teams.
The Galaxy's goal came after half an hour. Juninho let fly from 35 yards and his arcing shot took a slight deflection on the way into the top corner of the net.
The three points marked LA's first CCL win away from the Home Depot Center this year.
David Beckham wore the captain's armband for LA with Landon Donovan only fit enough to start on the bench. The American attacker entered in the final 20 minutes and skied a good chance from the top of the box near full time.
The other MLS sides to advance to the knockout rounds of the CCL were Seattle Sounders FC and Toronto FC. FC Dallas and the Colorado Rapids were eliminated. The three MLS sides will join Mexican clubs Morelia, Santos, Pumas and Santos, as well as El Salvador's Metapan in the quarters, which begin in March.