Vegas had outshot Montreal 39-17 when Fleury turned it over to Anderson, who was alone in front to score into a wide-open net for his first goal in 13 games.
"Those type of events are tough to recover from," Golden Knights coach Peter DeBoer said. "We talked between the third and the overtime about trying to get our mojo back. I didn't think we were poor in overtime, but there's no doubt that carried into the overtime for us a little on the negative side and for sure gave them some pop going out to overtime."
Cole Caufield scored, and Carey Price made 43 saves for the Canadiens, who played without coach Dominique Ducharme after he tested positive for COVID-19.
"We just don't quit," Price said. "... Not having Dom behind the bench is just another twist we're going to have to deal with."
Nicolas Roy and Alex Pietrangelo scored, and Fleury made 24 saves for the Golden Knights, who lost Game 2 3-2 at home on Wednesday.
"Fleury's played great for us all year," Vegas captain Mark Stone said. "It's one mistake. We had to bail him out and did not play the way we played for the first 60 (minutes) in overtime."
Game 4 is at Montreal on Sunday. Teams that take a 2-1 lead are 57-18 (.760) winning a best-of-7 Stanley Cup Playoff semifinal series.
Pietrangelo put the Golden Knights up 2-1 at 2:22 of the third period. The defenseman drove the middle to take a return pass from Max Pacioretty and scored his third goal in two games on a wrist shot from the top of the slot.
Vegas took the first 12 shots on its way to outshooting Montreal 17-3 in the first period and 30-8 through the second, but the game was tied.
"You guys all saw it tonight, it was incredible," Anderson said of Price. "Not only to play like that, but for the fans too. He kept us in the game all game, and we wouldn't be sitting here talking about this without him."