Philadelphia scored the game's final five goals to end Detroit's four-game winning streak.
The Red Wings owned a 3-1 lead with time running down in the second period, but Matt Read started the comeback by scoring with 3:22 left in the middle stanza.
Claude Giroux tied it on the power play 5:15 into the third, Couturier lifted the Flyers in front just over three minutes later, Scott Hartnell added a power-play goal and Couturier finished things off with an empty netter in the closing minute.
Steve Mason surrendered three goals during the game's opening 30 minutes, but was perfect the rest of the way and picked up the win with 32 saves.
Jimmy Howard allowed five goals on 33 shots.