Jaccob Slavin and Brady Skjei also scored for Carolina (42-12-8), which has won three straight and 15 of its past 18.
The Hurricanes moved four points ahead of the New Jersey Devils for first in the Metropolitan Division. The Devils lost 4-3 to the Toronto Maple Leafs on Tuesday.
"I don't know what is wrong with us, but we somehow always have a little sloppy start against the lower-ranked teams," said Carolina forward Jesperi Kotkaniemi, who won it in the sixth round of the shootout. "We have to get rid of that if we want to be the best in the League."
Antti Raanta allowed two goals on 10 shots before he sustained an undisclosed injury. He was replaced at the start of the second period by Frederik Andersen, who made 13 saves.
Hurricanes defenseman Jalen Chatfield left the game after the second after sustaining an upper-body injury.
"Precautionary really on both guys, so I don't think it will be serious for either one," Brind'Amour said.
Alex Belzile and Michael Pezzetta each had a goal and an assist, Chris Tierney had two assists, and Allen made 36 saves for Montreal (26-33-5) which had lost three in a row to conclude a four-game road trip.
"You've got to give a lot of credit to all these guys in this room. It's not an easy team to play to begin with," Allen said. "But the situation we're in, after coming off almost two weeks on the road, to come back from a West Coast trip, the first game's always really difficult [and] you don't have your legs.
"I was really impressed and proud of that group, the way they battled against a [heck] of a hockey team in a tough circumstance."