Vela scored on a penalty kick in the 86th minute, giving him 62 goals and 38 assists in 98 regular-season matches.
Sebastian Giovinco (95 games) and Robbie Keane (96 games) reached the mark faster than Vela.
The win is Austin's first in four meetings with LAFC and moved the teams into a tie atop with the Western Conference with 23 points. Austin FC already has seven victories in its first 12 matches after winning nine all last season.
LAFC appeared as if it was going to clear a corner kick attempt in the 13th from Fagundez, but Diego Palacios was unable to clear it. Palacios' pass was intercepted by Johan Valencia, who passed to Gabrielsen before he was able to fire a right-footed strike from eight yards out into the bottom left corner of the net.
It was Austin's fifth goal in the first 15 minutes of a match this season, which is tied with NYCFC for most in the league.
Fagundez hit the crossbar in the 23rd minute, but converted in the 80th minute. The midfielder got a great pass from Sebastian Driussi, went on a break up the left side of the field and buried it from 12 yards out to make it 2-0.
Vela cut the margin in half six minutes later when he drew a penalty kick after being fouled by Austin goalie Brad Stuver inside the box. Vela then put in a left-footed shot into the bottom left corner from the spot.
LAFC had an 18-9 advantage in shots and 7-2 in shots on goal, but dropped its second straight. Stuver made six saves for Austin, while Maxime Crepeau had none for LAFC.