Briere, who had scored 104 points in 103 playoff games leading into the game, blasted a shot from the point 4:36 into overtime to continue his outstanding postseason record.
"I can't explain it," Briere told reporters. "When I was young I always dreamed that one day I would have the chance to be there (in the playoffs).
"Now that I am there I just want to take every opportunity that I have every game that I can get in."
He did exactly that in the Flyers' first round victory over the Pittsburgh Penguins, scoring five goals and tallying three assists in six games.
Sunday's game-winner came two minutes after he had a power-play goal waved off when officials ruled he had kicked the puck into the net.
The right wing's first goal had leveled the score at 1-1 in the second period.
Philadelphia scored again 37 seconds later with James van Riemsdyk getting the goal.
Travis Zajac knotted the score 13:53 into the period and both teams scored in the third.
Claude Giroux collected his seventh goal of the playoffs for Philadelphia four minutes into the period and Petr Sykora delivered the equalizer at the 11:22 mark.
Zach Parise had New Jersey's first goal.
Flyers netminder Ilya Bryzgalov stopped 23 shots with Martin Brodeur making 32 saves for the Devils.
Game Two is Tuesday in Philadelphia. The two teams split six games in the regular season.
The third-seeded Coyotes have relied on overtime triumphs and heroics from goaltender Mike Smith to reach the second round for the first time in franchise history, but needed only a dominant second period to tighten their grip on the best-of-seven series.
Martin Hanzal, whose assist set up the OT game-winner in the series opener on Friday, led from the front with a goal in the second period in which Radim Vrbata and Taylor Pyatt also scored to put the home side up 4-2.
"We didn't totally rely on (Smith) but he was huge again for us," said Phoenix winger Shane Doan, who notched the game's final goal in the third period.
"We've had the lead in the third in a few games but haven't played the way we wanted to. Tonight was a little closer to that."
The Predators will try to claw back into their series beginning with Game Three on Wednesday in Nashville.
They briefly pulled within 4-3 on Ryan Suter's score early in the third but Doan quickly halted their momentum with his late tip-in.
Pekka Rinne made 34 saves for the Predators and Smith made 30 stops on a night he did not have to carry the load.