Jeff Petry scored, and Price made 28 saves for Montreal (11-6-7), which is 2-1-3 since coach Claude Julien was fired Feb. 24 and replaced by Dominique Ducharme. Montreal has a five-game point streak (2-0-3) and is 4-0-2 against Vancouver this season. They will play again here Wednesday.
"We're getting better and better," Ducharme said. "I don't think we gave them much. I think we gave ourselves enough chances to score goals, but we've got to finish. Defensively without the puck, the way we're coming back and applying pressure and forcing plays and creating turnovers, we're doing a good job. So, a lot of good things."
Montreal went ahead 1-0 at 4:37 of the first period on Petry's power-play point shot past Demko, who was being screened by Corey Perry. The Canadiens were 1-for-2 on the power play and are 5-for-11 (45.5 percent) in six games since Ducharme took over after failing to score on 10 chances in the previous six games.
"I think it's the compete level," Petry said of the power play, which is run by new assistant coach Alexandre Burrows. "We have to have that five-on-five mentality that if the puck is loose we have to outnumber them and hound the puck and work hard to get the puck back, and I think that has been ingrained in us since he's come in and we're executing it."
Price came into the game with an .898 save percentage, the lowest of his 14 seasons in the NHL, but has given up one goal in each of his past three starts while stopping 82 of 85 shots for a .965 save percentage.
"Just playing better. I'm not going to give you my technical secrets," Price said. "The guys are playing solid hockey in front of me and letting me see the puck."
Demko, who has allowed four goals while winning his past four starts with a .969 save percentage, stopped Joel Armia on a breakaway at 10:15 of the second period.
"It looked like Carey was playing really well, and when he's on his game he's tough to beat," Demko said. "So if they get another one, it's tougher to come back."
NOTES: Canucks defenseman Jordie Benn left 47 seconds into the third period after blocking a shot, but there was no update after the game. … Vancouver defenseman Quinn Hughes played a game- and NHL career-high 30:21. … Montreal is 0-4 in overtime and 0-3 in the shootout this season. … Canadiens equipment manager Pierre Gervais worked his 3,000th NHL game.