Bruce Arena's men won at the StubHub Centre on Monday thanks to Sean Franklin, but goals from Sebastian Velasquez and Chris Schuler ensured that the Claret and Cobalts' challenge for a second MLS Cup remains on track.
The home side took the lead at Rio Tinto Stadium in the 35th minute, when Velasquez headed home Chris Wingert's cross.
Neither team could find a winner before the 90 minutes were up, meaning that extra time - and possibly penalties - would be needed to separate them.
The decisive moment came in the 103rd minute, when Galaxy failed to deal with a Javier Morales corner, which allowed Schuler to prod home from close range.
A glorious chance to level the game fell to Franklin two minutes into added time in extra time, but he was unable to find the target with his header.