Brayden Schenn had the Flyers' other goal and Sergei Bobrovsky made 35 saves. Philadelphia is 18-3-3 against the Hurricanes since the start of the 2006-07 season, including 10-1-1 in Carolina.
Jerome Samson scored his first NHL goal for the Hurricanes, who were 4-0-1 in their previous five home games. Carolina has dropped four five overall.
Cam Ward made 35 saves for the Hurricanes, and the game was close throughout despite the disparity in records.
The Flyers entered the night four points behind the New York Rangers at the top of the Eastern Conference, while the Hurricanes were ahead of only the New York Islanders.
Philadelphia wanted to make amends for its performance Sunday against Ottawa. The Flyers blew a two-goal lead in that game and lost 6-4, and Daniel Briere described his team's effort as "soft" afterward.
With the score tied at 1 early in the third period, Couturier received the puck near the Carolina blue line along the right boards. He accelerated forward and beat Ward on the blocker side with a shot that stayed just inside the left post.
After a scoreless first period, Philadelphia opened the scoring early in the second. Ward originally stopped Wayne Simmonds' attempt on the rush, but Schenn was in front and cleaned up the rebound.
Carolina tied the score on the power play. Justin Faulk slapped a shot from the high slot, and the rebound came to Samson, recalled from the Hurricanes' minor league affiliate in Charlotte earlier Tuesday. Samson beat the outstretched Bobrovsky and had a wide grin on his face when he got back to the bench.