It was Brossoit's first shutout of the season. He played 28 games for the Oilers from 2015-18.
"I definitely was juiced up to play this one," Brossoit said. "I don't think I've won against Edmonton since moving on from them, so it felt nice to win, and to get the shutout, it meant even more."
Brett Howden scored for the fourth straight game, and William Karlsson and Reilly Smith each had a goal and assist for the Golden Knights (28-16-3), who have won three in a row.
"You have to give them credit, I thought they were real good the first 10 minutes, and I thought we kind of worked our way into the game," Vegas coach Peter DeBoer said. "Our best player early was our goalie, and I thought once we got our legs going and got up to speed, I liked a lot of things about the rest of the night."