WATCH: [All Flyers vs. Canadiens highlights]
"It's a good two points," Flyers coach Dave Hakstol said. "It's nice to get a shootout win and get the extra point. I think that's important right now from our standpoint. Every point is critical, and we just keep pushing."
Sean Couturier scored the deciding goal in the shootout for Philadelphia, which has won six straight and is 10-0-2 in its past 12 games.
The Flyers have not lost in regulation since Feb. 1 against the New Jersey Devils.
"We have to keep going, keep working hard, and play the way we played tonight," Mrazek said. "And if we're going to do that, I believe we can stay in first place for a long time."
The win was Philadelphia's seventh straight on the road, which is tied for the second-longest streak in its history. The Flyers record for consecutive road wins is eight from Dec. 22, 1982-Jan. 16, 1983.
Charlie Lindgren made 33 saves for Montreal (23-29-10), which has lost two straight (0-0-2) and eight of its past nine (1-4-4). The Canadiens are 13 points behind the Columbus Blue Jackets for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.