Sebastian Velasquez levelled the Western Conference semi-final tie at 1-1 with his first-half goal at Rio Tinto Stadium.
The Galaxy, who won the first leg 1-0 at home, were fortunate not to lose in regulation time after a Real Salt Lake goal was disallowed.
But Chris Schuler scored the winner on 102 minutes to see Real Salt Lake to their first post-season win since 2011.
Jason Kreis' men will face Portland in their conference final after the Timbers beat the Seattle Sounders 3-2 in the second leg and 5-3 on aggregate.
Velasquez's opener against the Galaxy ended a run of 370 minutes without a play-off goal for Real Salt Lake.
Chris Wingert's cross from the left found Velasquez unmarked in the area and the 22-year-old headed into the bottom corner.
And the hosts appeared to have scored the winner in the 84th minute through Javier Morales.
The Argentine whipped in a set-piece which went over the heads of everyone in the area, only for it to be ruled out for an apparent foul in the box.
Los Angeles looked troubled from the dead ball throughout the contest and it was no surprise when the winner came from a set-piece.
Galaxy defender Kofi Opare appeared to have Morales' free-kick covered but it floated over his head and Schuler slid in at the back post to send Real Salt Lake through.
In Portland, the Timbers held on against Seattle despite a nervous finish at Jeld-Wen Field.
A Will Johnson penalty and goals from Diego Valeri and Mamadou Danso sent Portland on their way to the conference final.
But two goals in three second-half minutes from DeAndre Yedlin and Eddie Johnson gave the Sounders hope, although they were unable to score another and the Timbers held on to maintain their maiden run in the play-offs.
