"Just can't quit on a game, right? I think we've been pretty good at that all year, is sticking around games and finding ways to get goals late, keep us alive, and tonight we found a way to get the next one," said Brown, who scored for the second straight game.
WATCH: [All Golden Knights vs. Kings highlights]
Jeff Carter scored his first goal of the season, and Jonathan Quick made 37 saves for the Kings, who defeated the Golden Knights for the first time (1-1-1).
Los Angeles (34-24-5, 73 points), which won for the first time in three games, is tied with the Calgary Flames for the second wild card into the Stanley Cup Playoffs from the Western Conference. The Kings and Flames trail the Anaheim Ducks by one point for third in the Pacific Division.
Erik Haula and Reilly Smith scored for the Golden Knights, who lost for the first time in three games. Marc-Andre Fleury made 41 saves.
Vegas (41-16-5, 87 points) leads the Nashville Predators by two points for first in the Western Conference.
The Kings rallied late by getting to the net. Brown was standing in the crease and was struck in his leg by Tyler Toffoli's long shot. He gathered the puck and reached around a sprawling Fleury for his second overtime goal this season.