Couturier got the winner at 8:28 of the third period by tipping in a shot.
Scott Hartnell and Matt Read had a goal and two assists and Steve Downie and Claude Giroux had a goal and an assist each for Philadelphia. Steve Mason stopped 32 shots.
Philadelphia finished 3 for 3 on the power play.
Tomas Tatar scored twice and center Johan Franzen also scored for Detroit. Jimmy Howard made 28 saves.
Giroux tied it at 3, with a power-play goal 5:15 into the third period. He got his fifth goal, on a shot from the top of the left circle.
Hartnell made it 5-3 with a power-play goal 9:58 into the third for his sixth goal. Couturier added an empty-net goal in the last minute.
Read scored from the front of the net with 3:22 remaining in the second to pull the Flyers within 3-2. It was his ninth goal.
Detroit scored twice in less than two minutes in the middle of the second period to go up 3-1.
Franzen beat Mason with a slap shot from the top of the left circle off the rush to break a tie at 8:24. It was Franzen's eighth goal.
Tatar got his second goal of the game on a 5-on-3 power play by putting in a loose puck from a goal-mouth scramble with 9:47 left in the middle period. It was Tatar's fifth goal.
The Red Wings' Justin Abdelkader hit the post on a partial breakaway with 1:32 left in the first.
Tatar's first goal of the game opened the scoring with 9:43 left in the opening period on a one-time shot from the bottom of the left circle.
Downie's power-play goal tied it with 6:12 remaining in the first period. He beat Howard with a wrist shot from the inside edge of the right circle. It was Downie's second goal.
NOTE: The Red Wings were without both of their star centers, Pavel Datsyuk and captain Henrik Zetterberg. Datsyuk missed his fifth game with a concussion and Zetterberg sat out his first game because of a herniated disc in his back. Zetterberg is expected to miss at least two weeks. ... Flyers' C Vincent Lecavalier missed his second game with back spasms. ... Detroit RW Todd Bertuzzi missed his sixth game with an upper-body injury. ... Philadelphia RW Adam Hall is from Kalamazoo and played at Michigan State. .. Red Wings D Danny Dekeyser, who has missed seven games and is on the long-term IR with a separated left shoulder, began skating on Wednesday morning. ... Detroit LW Justin Abdelkader went to the locker room late in the second period after being shoved into the boards from behind by Flyers' defenseman Luke Schenn, who got a charging penalty. Abdelkader returned in the third.