Tokarski stopped three Panthers in the shootout, clinching the victory when he stopped Jonathan Huberdeau's wrist shot in the bottom of the third round.
"It was unbelievable," Canadiens forward Lars Eller said of Tokarski's game. "It was like having [Price] in there. He did exactly what he was supposed to do and a little bit more. All the boys were really happy for him."
Price started the previous 11 games and allowed nine goals in his past seven.
Tokarski was making his first start since Nov. 29. His performance was no surprise to Florida coach Gerard Gallant, an assistant with Montreal the past two seasons.
"I saw Tokarski last year and he was real good," Gallant said. "He's a solid guy and real good goaltender, and tonight he looked like Carey Price. Everything stayed on him. There weren't many rebounds. He played an outstanding game for them."
Parenteau beat goalie Roberto Luongo in the third round with a wrist shot and is 4-for-6 in the shootout this season. All four goals have been game-deciders.
"He's got that magic touch in that shootout," Montreal coach Michel Therrien said. "That was a huge point, a huge point when you get to the end of the season. He's certainly got that magic touch."
Parenteau said, "I think that's my fourth (shootout winner), but who's counting? I was trying to go high glove and I went low glove. I missed my shot, really, to be honest with you guys."
Brendan Gallagher scored with 1:06 left in the second period for Montreal (24-11-2), which has won four in a row and seven of eight. Montreal is 14-0-0 when scoring first.
The game drew a sellout crowd of 19,614, the largest attendance of the season at BB&T Center, with a large portion rooting for the Canadiens.
There was a loud cheer when Gallagher scored and when the Canadiens won the game.
"I was pretty surprised actually," Gallagher said of the reaction after his goal. "I was trying to figure out if it went in or not just because [of] how they were cheering. I was a little confused, but it was nice to hear."
Florida (16-9-9) has a point in its past eight home games, 6-0-2 since a 4-1 loss to the Minnesota Wild on Nov. 24.
Nick Bjugstad scored with 1:49 left in the third period for the Panthers; Luongo made 33 saves and is 1-7-0 in his past eight starts against the Canadiens.
"It was a good hockey game," Gallant said. "Two teams battled real hard and there were some good chances both ways, great goaltending. We got a big point, but when you get that close, you want that second one."
Florida missed a chance for its first three-game winning streak of the season. It has won two in a row four times.
The best scoring chance of overtime came in the final minute after Tokarski turned over the puck and Jimmy Hayes had an open net from the bottom of the left circle, but his long-range shot went wide.
Bjugstad tied the game when he took a centering feed from rookie defenseman Aaron Ekblad and beat Tokarski with a wrist shot from the slot.
"I think he had two guys on him; I was close to changing actually," Bjugstad said. "I didn't really think much of the play. Had confidence in [Ekblad], though, and he threw it right to me. It was a good overall play by him. That's why he was the first overall pick, he can make plays like that."
Gallagher scored on Montreal's 20th and final shot of the second period, a wrist shot from outside the right dot that trickled through Luongo's legs. It was Gallagher's fifth goal in 10 games.
"I was really just trying to put it on net," Gallagher said. "I shot it through a screen, so I knew it would be a little tougher to pick up. He had made some pretty big saves earlier, so it was nice to see one squeak through."
Luongo had been spectacular in the second period. He stopped Jiri Sekac on a breakaway in the first minute, blocked a cross-ice pass to thwart a 3-on-1, and had quick reactions to stop a couple of deflections, including one on a slap shot that shattered the stick of Panthers defenseman Willie Mitchell.
Aleksander Barkov came close to giving the Panthers the lead in the second period. He put the puck in the net off a scrum in front of Tokarski, but the goal was waved off because it was ruled Tokarski was shoved into the net.
Barkov hit the crossbar with a shot from the slot about two minutes before Gallagher scored.
Huberdeau's 14-year-old sister, Josiane, sang "O Canada" before the game. She was among approximately 85 friends and family members cheering on Huberdeau, whose hometown is the Montreal suburb of Saint-Jerome.
"It was a good moment," said Huberdeau, who had an assist on Bjugstad's goal. "I'm proud of her and it was just nice to see her on the ice too."