Alvaro Saborio had the first effort on goal in the third minute of the game, but it was the Galaxy who made the stronger start, with Robbie Keane and Gyasi Zardes both shooting wide inside the first 12 minutes.
The hosts finally got their reward two minutes after the restart, and it took a stunning long-range effort from Sean Franklin for them to get their noses in front.
Donovan nearly doubled their advantage moments later after getting on the end of Keane's cross.
Devon Sandoval was presented with an opportunity a minute after coming on for Saborio, but he could not connect with a header at full stretch.
However, Bruce Arena's men should have gone 2-0 up when Keane fired wide after being slipped through on goal with six minutes remaining.