Carey Price was brilliant Thursday, showing why he will be in the Hockey Hall of Fame someday.

But even his greatness couldn’t hold off the relentless Flyers.

Sean Couturier scored with 4 minutes, 5 seconds left in overtime as his shot trickled over the goal line after hitting off Price, lifting the Flyers over Montreal, 3-2, at the Wells Fargo Center.

It gave the Flyers points in four straight games (3-0-1) heading into the weekend’s daunting task: back-to-back games at Toronto on Saturday and at Boston on Sunday.

The Flyers (8-5-2) have 18 points, their most in the first 15 games since the 2011-12 team started 8-4-3 for 19 points.

Carter Hart (22 saves) defeated his boyhood idol, Price, who kept his team in the game by making 40 saves on 42 shots through regulation. The Flyers took a 2-1 lead into the third, but it could have easily been at least 4-1 if not for Price’s sensational play.

“I think Price is the only reason why they got a point tonight,” said Hart, whose team outshot the Canadiens, 43-24. " ... I thought we really competed hard tonight."

Hart admitted that defeating Price made the victory extra special.

“It’s the first time to beat him, so that was pretty cool,” he said. “The first time I played against him it didn’t go so well. I was a little star-struck and got pulled in that game” last season.

Couturier’s fifth goal of the season, which went through the legs of defenseman Victor Mete, ended things.

“I was just trying to use their D-man as a screen,” Couturier said after whipping a shot from the high slot past Price for the fourth overtime goal of his career. “I had a lot of room. There was probably a bad gap there. T.K. [Travis Konecny] was probably using his feet and backing up guys and dishing it to me. I tried to use a quick release and surprise Price. He’s a good goalie when he sees the puck, and I think that’s the way you have to beat him -- surprise him."

With the score tied at 2-all and 6:43 left in the third, Flyers rookie Carsen Twarynski was pulled down by defenseman Shea Weber and awarded a penalty shot. But Twarynski was denied by Price.

A misplay behind the net by Hart led to a goal by Weber that tied the score at 2-2 with 15:55 left in regulation.

A delay-of-game penalty on Montreal gave the Flyers their sixth power play with 9:27 left in regulation, but they did little against the NHL’s second-worst penalty kill. That made them 0-for-6 on the power play, though they scored one goal just after a Canadien came out of the penalty box.

Montreal did not have any power plays on the night.

Earlier, shortly after a Flyers power play expired, James van Riemsdyk scored on a rebound after the Flyers swarmed the net and created a scramble in front of Price. It was van Riemsdyk’s fourth goal – and eighth point in the last eight games -- and it put the Flyers ahead, 2-0, after 49 seconds of the second period.

Montreal was getting totally outplayed, getting outshot by a wide margin, and getting beat to loose pucks for most of the first two periods. But the Habs got to within 2-1 on a rare trip into the Flyers’ end.

Defenseman Ben Chiarot scored on a rebound of a shot taken by ex-Flyer Nick Cousins with 6:16 left in the second period. Phillip Danault got past a failed Jake Voracek poke-check in the neutral zone to put the sequence in motion.

At one point in the second, the Flyers had a 33-13 shots advantage – and a 32-8 domination since the game’s first five-plus minutes.

With 6:09 left in the first, they took a 1-0 lead when defenseman Phil Myers scored on a wrist shot from the point. Konecny and Twarynski screened Price. Twarynski kept the play alive with some good work behind the net.

“I was just trying to play aggressive on their forward and trying to keep the puck in,” Myers said after his first goal of the season and second in his young NHL career. “I tried to keep it simple and put it on net.”

Price made 17 saves in the first, none better than his robbery of an all-alone Couturier in front with 5:09 left in the session.

Couturier, who did not take any faceoffs because of a strained shoulder, later had the final say.