Jakub Voracek, Matt Read and Brayden Schenn also scored for Philadelphia, which went 3-0-1 on a four-game homestand.
Joe Corvo scored the tying goal in the third period, and Jiri Tlusty and Jeff Skinner also had goals for the Hurricanes, who had won two in a row.
Briere skated down the left side, made a move around the circle and fired a shot that hit the far post and trickled past goalie Cam Ward. The game went to overtime because Ward stoned Read on a point-blank shot at the end of regulation.
The Flyers have gone 24-4-4 against the Hurricanes since Nov. 28, 2003. Philadelphia is 5-6-1 at the quarter point of the lockout-shortened 48-game schedule. Carolina fell to 5-4-1.
Ilya Bryzgalov made 30 saves in the victory and earned his third assist of the season.
Corvo scored his first goal of the season 7:58 into the third period to tie it at 3. After a flurry of shots on Bryzgalov, Corvo ripped a slap shot that appeared to bounce off Flyers defenseman Kurtis Foster and behind Bryzgalov.
Voracek put the Flyers up 3-2 only 15 seconds after Skinner's goal tied it in the second. Schenn took an outlet pass from Bryzgalov, skated down the left side, and made a perfect centering pass to Voracek, who beat Ward with a backhand redirection for his third goal.
Skinner scored off an outstanding individual effort. His shot from the left side was stopped by Bryzgalov, but he skated behind the net, spun around, and lifted a shot over Bryzgalov, who was slow going side to side, for his sixth goal.
Read gave the Flyers a 2-1 lead 93 seconds into the period when he scored his fifth goal and first against a team other than Florida. After a turnover by Eric Staal, Mike Knuble passed the puck to Read, who wristed a shot from inside the right circle over Ward's glove and just under the crossbar.
After failing to score during a 5-on-3 advantage, the Flyers tied it at 1 on an even-strength goal by Schenn in the first. He deflected Kimmo Timonen's shot from the left point past Ward.
A five-minute major on Carolina's Tim Wallace for boarding followed by a tripping call on Bobby Sanguinetti gave Philadelphia a two-man advantage, but they couldn't get a shot past Ward. Timonen and Claude Giroux each hit the post on Philadelphia's two best chances.
Tlusty gave the Hurricanes a 1-0 lead just 17 seconds into the game, putting in a rebound for his first goal of the season. Corvo ripped a slap shot from the top of the right circle that Bryzgalov kicked away. The puck bounced out to Tlusty, who lifted it into the top left corner.