Justin Peters stopped 28 shots and earned his first win of the season while starting his fifth straight game since replacing the injured Cam Ward on Oct. 24.
Scott Hartnell had given Philadelphia a 1-0 lead with 3:58 left in regulation, but Jared Staal tied it with 52.3 seconds remaining to force overtime.
The Flyers, who have lost three of four, have scored more than two goals only once this season in 14 games.
Carolina was 0-of-4 on the power play, including a 5-on-3 advantage that carried over into the third period.
Philadelphia, which has the 29th ranked power play in the NHL, went 1 for 4 with a rare man-advantage goal. Steve Mason made a season-high 36 saves on 38 shots, also a season high.