Brayden Schenn notched the other goal for the Flyers, who improved to 9-1-0 in their last 10 games in Raleigh.
Jerome Samson provided the sole offense for the Hurricanes, who have lost four of five.
Cam Ward made 35 saves in defeat.
Couturier retrieved a puck kept in the Carolina zone by Harry Zolnierczyk, moved up the right wing and beat Ward high over his left shoulder from the top of the circle at the 3:23 mark of the third period for a 2-1 Philly edge.
Ward was sent to the bench for an extra skater with 1:25 remaining. The Flyers missed several chances with the empty net, and endured an icing call with 2.1 seconds left.
But the ensuing faceoff was pushed into the left corner as time expired and the visitors came away with their 11th win here since 2005.
After a scoreless first period which featured multiple close calls, Schenn put the Flyers on the board at 4:20 of the second, when he followed up a Wayne Simmonds shot and poked home the rebound in front.
It was the only blemish on Ward in an 18-shot period for Philadelphia.
Samson then shoveled home a Justin Faulk rebound to tie the game on a power play at 12:37.
Laviolette upped his record to 100-59-21 with the Flyers. He became the eighth man in franchise history to attain the mark, following Fred Shero, Pat Quinn, Mike Keenan, Paul Holmgren, Terry Murray, Ken Hitchcock and John Stevens...Carolina defenseman Jaroslav Spacek left the game with a lower-body injury late in the first period and did not return.