Price stopped Tuomo Ruutu, Jussi Jokinen, Rod Brind'Amour, Sergei Samsonov and Bryan Rodney before Lapierre finally beat Carolina's Manny Legace on Montreal's sixth attempt. Price then got enough of Matt Cullen's shot off a deke on the Hurricanes' next attempt to leave the puck on the goal line to the delight of the sell-out crowd.
"I think it hit the last part of the steel on my skate," Price said. "It hit me right in the toenail."
As pleased as he was to deliver the shootout winner, Lapierre suggested that Price's final save capped the evening.
"I would say the last shot with the puck on the red line, spinning, was pretty exciting," Lapierre said. "I just tried to keep my speed the whole way. That was the plan, go fast and shoot high, and I had him."
Price, who tied a team record with 53 saves Saturday, stopped 30 shots, including consecutive saves on Brandon Sutter and Erik Cole during the waning seconds of overtime while Carolina enjoyed a 4-on-3 advantage.
Andrei Kostitsyn drew Montreal even 17:14 into the third. Kostitsyn chased down his own rebound behind the Hurricanes' net and banked a shot in off Legace's skate from behind the goal line for his second goal of the season.
Max Pacioretty also scored for the Canadiens, who ended a three-game losing streak at home.
"We didn't play the way we wanted in the first two periods but we got the two points, that's the only thing that matters right now," Lapierre said.
Sutter and Jokinen scored 2:52 apart in the second period as Carolina came within 2:46 of recording its second regulation win of the season.
"That's the way hockey is," Sutter said. "It's not very often that you get up by a goal and you hold them off all game. We had a chance to go up by a couple, we didn't, and they ended up coming back."
The Hurricanes ending a franchise-record skid at 14 games with a 5-4 shootout win over Minnesota on Sunday.
Michael Leighton made 10 saves before leaving 7:12 into the second period because of a lower-body injury. Legace stopped 23 shots the rest of the way.
"Manny did great in net there and obviously shootouts can go either way, so it's a tough one," Sutter said.
Aaron Ward returned to the Hurricanes lineup after missing Sunday's game because of a lower-body injury.
Pacioretty opened the scoring 5:41 in when he took Tomas Plekanec's centering pass from the left side and put a shot from the slot past Leighton's right pad.
Sutter scored his fifth of the season 3:28 into the second to draw Carolina even. Jokinen got his fifth at 6:20 to put the Hurricanes up 2-1.
Cullen intercepted Spacek's clearing pass in the neutral zone on the play that led to Jokinen's go-ahead goal.
- F Stephane Yelle was back in the Hurricanes lineup after leaving Sunday's game with an undisclosed injury.
- Legace stopped Mike Cammalleri, Scott Gomez, Kostitsyn, Plekanec and Guillaume Latendresse in the shootout.