LAFC (9-9-6, 33 points) won its third straight game, swept the three-game season series from expansion Austin and jumped above the West's playoff line.
Austin (5-15-4, 19 points) dropped its fourth consecutive match and remains at the bottom of the West standings.
Los Angeles goalie Tomas Romero did not have to record a save to earn the victory.
Austin outpossessed LAFC 51.1 percent to 49.9 percent in the scoreless first half, but the visitors produced most of the offense, racking up eight shots to just three for Austin.
Los Angeles recorded the lone shot on goal over the first 45 minutes, a wicked attempt from Cristian Arango in the 24th minute that was stopped by Austin keeper Brad Stuver with a desperate dive to his right.
LAFC pressed the action early in the second half, pumping out three shots in the first 10 minutes. That pressure eventually led to success as Arango produced the game's first goal on a 59th-minute penalty kick. The goal came after the video assistant referee ruled that Austin's Diego Fagundez had committed a handball in the penalty area.
Fagundez made up for his penalty by pulling Austin even in the 67th minute. Fagundez was in the correct spot for the rebound after Cecilio Dominguez's shot hit the right post and bounced to the front of the goalmouth.
Musovski then hit the game-winner, streaking down the field between two defenders to get to a through ball from Jose Cifuentes.
LAFC beat expansion Austin FC 2-0 at home the season opener on April 17 and by the same score in the Texas capital on July 7.
Austin still has the fewest goals in the league, with 22 goals through 24 games. Its 15 losses are tied with Toronto for the most in MLS.