Pacioretty earned the primary assist on all three of Galchenyuk's goals.
"I think he owes me dinner," said Pacioretty, who recently inherited linemates in Galchenyuk and Brendan Gallagher. "It's fun playing with those two. I'm not old by any means, but they bring my age down a little bit. I feel like I have a lot more energy playing with those guys at times.
"They're always going, always buzzing."
Pacioretty used to play on the first line with David Desharnais and Pierre-Alexandre Parenteau. Galchenyuk and Gallagher were on the second line with Tomas Plekanec.
The new moves paid dividends and sent the Hurricanes to their sixth straight loss.
Galchenyuk, Pacioretty and Gallagher peppered goalie Cam Ward with eight shots. Each forward finished with a plus-3 rating.
"We had good chemistry," said Galchenyuk, who had never had a multi-goal game. "Pacioretty made great plays on all three. I have to get used to his speed, and he has to get used to playing with me.
"My linemates made it really easy for me. We had a lot of fun out there. It took me two-and-a-half years to score two goals. It's even better to get three."
Brandon Prust added a goal, and Carey Price stopped 25 shots for the Canadiens (20-10-2).
Victor Rask was the only goal scorer for Carolina (8-19-3). Ward made 25 saves. The Hurricanes have scored just one goal in each of the losses in their skid.
"It's a tough league to win in when you're scoring one goal a game," said Hurricanes forward Jeff Skinner, who had a point and a minus-two rating. "We haven't been executing in the offensive end to earn a win."
After losing three straight on the road, the Canadiens have won three consecutive on their homestand.
Montreal scored the eventual winning goal at 8:33 of the second period with a highlight-reel effort involving four players.
Galchenyuk scored off a tic-tac-toe pass from Pacioretty on a sequence initiated by Gallagher and Andrei Markov. Ward was far from the crease when the puck entered the net.
"He's a young guy with a lot of potential," Canadiens coach Michel Therrien said of Galchenyuk. "We're all aware of that. Games like this one give him confidence. He's creating good chemistry with Pacioretty, and he's already very used to playing with Gallagher."
The Hurricanes have been outscored 18-6 during their losing streak. Carolina has scored more than two goals once in nine games.
Ward was way out of his net when Prust made it 1-0 at 9:15 of the first. David Desharnais faked a wraparound and lured Ward from the crease before dropping the puck back in front to Prust, who scored his second goal of the season.
Carolina came into the game having taken an NHL-low 76 penalties, but the Hurricanes took five against Montreal.
With Carolina trailing 2-0, Rask gave the Hurricanes life early in the third period.
Skinner's shot from the circle handcuffed Price, went off his shoulder, and trickled down to the goal line. Rask was there to nudge Price's pad, poking the puck into the net in the process.
Still a one-goal game, Elias Lindholm hit the post with five minutes remaining.
"That was a key moment in the game," Hurricanes coach Bill Peters said. "From that point, it gets extended and it looks like a one-sided game."
Galchenyuk put the game away after Pacioretty found him calling for a pass at the blue line. Galchenyuk went in on a breakaway and scored his eighth goal of the season at 16:41. The 20-year-old Russian completed the hat trick 30 seconds later.
Galchenyuk's linemates razzed him and gave him the cold shoulder.
"Pacioretty and Gallagher talked to each other to not celebrate with me," Galchenyuk said. "I was just skating by myself, and they were celebrating with the other players. All I could do was laugh."
Price is 7-2-1 in his last 10 home games against the Hurricanes.
NOTES: Montreal's Bryan Allen (flu), Lars Eller (upper body) and Mike Weaver (concussion) sat out. ... Carolina top-scorer Jiri Tlusty missed his second consecutive game with an upper-body injury. ... The Canadiens have won seven of eight at home against Carolina.