The Galaxy required three points after Morelia had gone top with a 2-1 victory over Alajuelense earlier in the week and they produced what they needed to move into the next phase of the competition.
Juninho's 29th minute proved the difference as the Galaxy ended second in Group A on the 12 points - the same as both Morelia and Alajuelense - but with a superior goal difference to the Costa Ricans.
LA skipper Landon Donovan was delighted with the result and was full of praise for his teammates for doing the job when required.
"I'm really proud of our guys," said Donovan. "We knew that this tournament was going to be difficult because a lot gets thrown at you, but we're fortunate.
"We've got a lot of guys who have been through a lot of these games and know what to expect.
"We did a pretty good job [at the end of the match]," he added. "Our center backs were great dealing with (Jerry) Bengston (Honduras international), who was a handful all night. And I think after about 15 or 20 minutes in the 2nd half, we kind of settled down, were steady passing.
"We ended up actually getting some good chances to score, but we did a good job seeing out the game."