The Flyers were 1-11 when trailing after two periods, a big blemish for a team that made edge-of-your-seat rallies almost routine in four seasons under coach Peter Laviolette.
They got the clutch goals they needed to keep the eighth and final playoff spot in the Eastern Conference within striking distance with their first three-game winning streak of the season.
"We don't have any choice," Laviolette said. "We've just been talking about our games and less on what other teams are doing around the league. The only thing we control right now is our games and we've got to make sure we win them."
With Philadelphia trailing 3-2, Simmonds tipped in the puck from the crease for the tying goal. Less than 2 minutes later, Gustafsson's pass for Matt Read went off a defenseman and into the net past a stunned Carey Price for his 10th goal and a 4-3 lead.
"We called ourselves the comeback kids last year and we are kind of back on that same page," Read said. "You play with confidence, and when you're down one or two goals you just fight back and it's another comeback. The comeback kids win again tonight."
"We just didn't match their intensity, it's just that simple," Montreal coach Michel Therrien said.
Prust and Desharnais scored 20 seconds apart late in the second period for Montreal, which is chasing Pittsburgh for the top spot in the Eastern Conference.
Prust evened the score at 2-all with his fifth goal of the season.
The Canadiens went ahead on a bit of an odd play. Desharnais' slapper from the circle knocked off Philadelphia's Brayden Schenn's stick and knuckled toward goalie Ilya Bryzgalov. Maybe the Flyers' much-maligned goalie lost the puck in the lights, but he dropped to his knees and ducked as the puck sailed over his head for the go-ahead goal.
He was mocked on social media by Flyers fans fed up with his erratic play over the last two seasons.
Bryzgalov signed a $51 million, nine-year deal before the 2011 season and has rarely played like the goalie expected to lead the Flyers to their first Stanley Cup since 1975. Before Wednesday's trade deadline, the Flyers acquired backup goalie Steve Mason from the Columbus Blue Jackets, a sign that perhaps the Flyers could use their amnesty buyout on Bryzgalov in the offseason and go with a cheaper and just as productive alternative.
Bryzgalov was bailed out late in the third when Simmonds caught a break and tipped in the puck for the tying goal.
Bryzgalov, making his 20th straight start, stopped 14 shots. He did not talk to the media.
Emelin's shot from the point through teammate's Brendan Gallagher's screen gave Montreal a 1-0 lead in the first.
The Flyers tied it when the puck shot off Montreal defenseman P.K. Subban's skate and into the net. Couturier was credited with his third goal. Gagne made it 2-1 with his third goal.
"They got four pretty lucky goals, in my opinion," Price said. "That's the difference."
With Pittsburgh's 6-1 loss to the New York Rangers, the Canadiens lost a valuable chance to gain ground in the standings.
"It would have been nice to be able to capitalize," Price said. "But that's the way it goes."
Flyers LW Zac Rinaldo left with a lower-body injury. ... The Flyers claimed C Adam Hall off waivers from Tampa Bay and recalled G Cal Heeter from their American Hockey League affiliate, the Adirondack Phantoms. ... Flyers D Chris Pronger, who hasn't played in more than two years because of lingering effects of a concussion, watched the game in a suite with team management.
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PHILADELPHIA, PA - APRIL 03: Wayne Simmonds #17 of the Philadelphia Flyers warms up prior to a game against the...