Owen Tippett also scored for Philadelphia (21-14-6), and Couturier scored the only goal of the shootout. Samuel Ersson made 17 saves and stopped three shots in the shootout.
Sean Monahan and David Savard scored for Montreal (17-17-6). Cayden Primeau made 37 saves, and another two in the shootout.
"Talked about it before the game, come out the right way, be ready to play," Canadiens forward Josh Anderson said. "We knew that they were going to be ready to play. They were. You're not going to win too many games when you play 40, 45 minutes of hockey. We made a good push at the end, but I thought we could be a lot better today."
The Flyers tied it 2-2 at 8:30 of the second period on Frost's power-play goal. Drysdale passed the puck to Frost, and the forward's shot from the top of the right circle got past Primeau.
"I think I was getting up the ice pretty good, trying to create as much as I can," Drysdale said. "Things I can work on through video and stuff like that, but overall just getting up and down the ice. Tried to have some good gaps."
It was the third straight game Philadelphia scored a power-play goal, the first time it's done that since Dec. 3-7, 2022. It entered the game No. 31 in the NHL on the power play this season (10.8 percent).
"We're starting to find a little more chemistry and create some plays and chances, guys know where they are," Couturier said. "It's a lot more fun these days than it was earlier in the year. I think we can still build it and be better for sure."
Couturier scored in the first round of the shootout, beating Primeau between his pads.
Monahan put the Canadiens ahead 1-0 at 1:29 of the first period when he tipped a point shot by Justin Barron past Ersson.
Montreal made it 2-0 at 11:28 when Savard's shot from the right point deflected off the stick of Frost and over Ersson's shoulder.
Those were the only two shots of the first period for Montreal.
"We need to protect that lead better," Canadiens captain Nick Suzuki said. "We were up two but we weren't playing good. Should make everyone feel a lot better about themselves and with our game, but you just kind of let them back in. Just wasn't good enough tonight. We need to be a lot better."