With his team leading 2-1 late in the third period, Fleury collected a Canadiens dump-in behind his own net but accidentally sent the puck into his skates, kicking it to open ice -- and leaving the net wide open for Josh Anderson to stuff in a score-tying goal.
Anderson also scored in overtime to give host Montreal a 3-2 win and a 2-1 semifinal series lead over heavily favored Vegas.
"It was an unfortunate bounce. There's nothing you can do about it," Golden Knights captain Mark Stone said. "Fleury has been great all year. It was one mistake, and we needed to bail him out and we didn't."
The gaffe was reminiscent of another infamous Fleury play. In the 2004 world junior championships gold-medal game against the United States, Fleury, playing for Canada, attempted to clear the puck but accidentally banked it off teammate Braydon Coburn, setting up Patrick O'Sullivan for the game winner for Team USA.
On Friday, Fleury smashed his stick on the ice in frustration after his gaffe. His teammates, though, were quick to defend the 17-year veteran.
"If we capitalize more in regulation, we're not even talking about that," Stone said.
Added veteran forward Reilly Smith: "Goals like that are going to happen with bad bounces. We just have to move forward. We should've done a better job in overtime to close it out."
However, coach Peter DeBoer said he believed the late mistake weighed on his team -- including into the extra period.
"Those type of events are tough to recover from," DeBoer said. "I didn't think we were poor in overtime, but there's no doubt that carried over."
Fleury, who earned his first Vezina Trophy finalist nomination this season, has a .921 save percentage through 15 starts this postseason. He has 90 career playoff wins, which places him fourth among goaltenders all time.
Vegas dominated a lot of Game 3, including a 30-8 edge in shots after two periods. In fact, DeBoer was quite happy with his team's performance, outside of the result.
"I loved our game," he said. "If we can play like that for the next week, I have a hard time believing they're going to beat us two more times."
Game 4 is Sunday in Montreal.