Couturier scored in regulation and had the only goal in the shootout on a nifty backhand shot, nearly identical to the one used by Forsberg so many years ago.
In the shootout, Couturier skated down the center of the ice and as he closed on in the net drifted left to draw the goaltender to the side of the crease, and tucked a backhanded shot into the net.
Forsberg popularized the move after using it in a shootout in the Olympics to win the gold medal.
''Everyone has seen those goals (by Forsberg),'' Couturier said. ''We have a lot of fun at the end of practice doing shootouts and that's one move I didn't do on him so maybe I caught him off guard. I did that in practice before, but never in a game.''
Oskar Lindblom and Joel Farabee also scored for Philadelphia. Carter Hart had 23 saves.
Wayne Simmonds, Sami Vatanen and Taylor Hall scored for the Devils. Mackenzie Blackwood had 28 saves.
Farabee's first career NHL goal at 8:31 of the third period tied the game and forced the extra period. The goal came after Michael Raffl's wrap-around shot bounced off Blackwood and out to Farabee on the point.
''It definitely felt good,'' Farabee said. ''I was pressing pretty hard lately so to get the monkey off my back felt pretty good.''
Farabee nearly won the game in overtime, but his rebound off his own shot was kicked in. Blackwood also had glove save on Claude Giroux to end overtime.
Hall scored a goal midway through the third period to give the Devils a 3-2 lead. Hall managed to slide a shot past Hart as Shayne Gostibehere moved past and shielded a shot as he was pushed from behind and into the goalpost by Justin Braun.
''I know it happened very, very quickly and my head was in the ice but anytime there's the puck is there you just get your stick going north and see what happens,'' Hall said. ''But it was nice to see that go in. It hasn't been a great year for me goal-scoring wise and any time you can get a goal, especially a goal that was harder, those kind of mean a lot.''
Simmonds opened the scoring with a power-play goal against his former team 6:24 into the first off a pass from Jack Hughes.
Midway through the period, the Devils turned the puck over at center ice and Lindblom scored for the Flyers, with the assists going to Travis Konecny and Sean Couturier.
The Devils appeared to take a 2-1 lead later in the period when Nico Hischier put a loose puck into the net on a rebound, but the goal was waved off.
''They said because we shot it that went under him and that was the difference as opposed if their guy shot it into them,'' Devils coach John Hynes said, adding that the referees told him the play couldn't be reviewed.
Hughes, the No. 1 overall pick in the NHL Draft, played just four minutes in the first before going to the locker room after he was hit hard into the end-boards by Flyers defenseman Matt Niskanen. Kyle Palmieri came to the defense of his teammate, picking up 17 minutes in penalties (two for instigating, five for fighting and 10 for misconduct), while Niskanen picked up a five-minute penalty for fighting.
Hughes returned at the start of the second period after going through concussion protocol.
The Flyers took a 2-1 advantage at the 8:26 into the second after the Devils failed to clear the zone and Couturier fired in a shot that bounced off of Devils defenseman Matt Tennyson and into the goal.
The Devils answered with 5:32 left in the second. Nico Hischier passed to Sami Vatanen at the center point and he scored past Hart on the power play as Giroux served a tripping penalty.
NOTES: Devils acquired goaltender Louis Domingue from Tampa Bay for a conditional seventh-round pick in the 2021 NHL draft. ... Hart shut out the Devils 4-0 on Oct. 9, stopping 25 shots in Philadelphia. ... C German Rubtsov, the Flyers' first-round pick in 2016, made his NHL debut. ... Simmonds had three assists in the last game vs. Flyers. ... Vatanen had four goals in 50 games last year and four already this season. ... The Flyers called up three players from the minors in D Phil Myers, C German Rubtsov, and F Carsen Twarynski from the AHL's Phantoms, placing F Scott Laughton on long-term injured reserve and sending C Misha Vorobyev and D Samuel Morin to Lehigh Valley.
Flyers: Host Toronto Saturday night.
Devils: At Carolina on Saturday night to start five-game trip.
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