Bruce Arena's Galaxy side needed a 52nd-minute goal from Marcelo Sarvas to edge the Seattle Sounders 1-0 at the StubHub Center.
It was a win the hosts needed in the Western Conference Finals heading into the return leg in a week's time.
Robbie Keane volleyed an early chance wide and Stefan Frei had to palm a Gyasi Zardes shot around his post as the Galaxy made a bright start.
The opportunities continued to come for the Galaxy and Keane lifted a close-range effort over the bar in the seventh minute.
Robbie Rogers almost put the hosts ahead on the stroke of half-time, but Frei came up with another save to deny him.
Sarvas finally put the Galaxy in front seven minutes into the second half.
AJ DeLaGarza found space down the right before passing to Sarvas, whose strike from inside the area took a deflection to beat Frei.
Both teams had chances after the goal, but the Galaxy claimed a deserved win.
In the Eastern Conference, the Revolution are even better placed after a 2-1 victory at the New York Red Bulls.
United States international Jermaine Jones scored an 86th-minute winner for the Revolution at Red Bull Arena.
Teal Bunbury put the Revolution ahead in the 17th minute, cutting inside from the right before curling a fine 20-yard effort into the corner with his left.
The hosts responded just 10 minutes later through the league's top goalscorer Bradley Wright-Phillips.
Peguy Luyindula had an effort saved before hitting the crossbar, but Wright-Phillips was on hand to nod home from close range.
But Jones tapped in the winner after Lee Nguyen fed Bunbury to square for the midfielder to make it 2-1.
