Valtteri Filppula, Claude Giroux and Matt Read also scored as the Flyers staved off elimination and now trail 3-2 in the best-of-seven series. Game 6 will be Sunday afternoon in Philadelphia.
Flyers goaltender Michal Neuvirth stopped 30 shots in the win.
Bryan Rust and Jake Guentzel each scored for the Penguins. Pittsburgh goaltender Matt Murray made 21 saves.
Couturier scored the game winner with a wrist shot from just inside the blue line. The puck appeared to deflect off Pittsburgh's Brian Dumoulin and past Murray.
Neuvirth made a spectacular save on Sidney Crosby with 1:01 remaining and Read sealed the victory with an empty net goal with 17.3 seconds left.
The Penguins struggled on the power play, going 0-for-5.
Pittsburgh's Conor Sheary attempted to tie the game but ripped a shot off the post with 11:20 left in the second. Sheary also had another shot go wide before clanging that one off the post.
The Penguins kept peppering Neuvirth with shots and eventually tied the game at 1 with 8:00 remaining as Rust scored on a wraparound.
The Flyers responded with several good chances in front of the net but couldn't capitalize. Giroux had the best opportunity but was stymied by Murray.
Pittsburgh went ahead 2-1 with 3:15 left in the second as Guentzel shot one through Neuvirth's pads.
The Flyers tied the game at 2 with a short-handed goal by Filppula with 1:45 left.
Giroux put the Flyers on the board for a 1-0 lead when he blasted a slapshot past Murray with 2:31 remaining in the first. It was Giroux's first goal of the series.
Pittsburgh's Phil Kessel nearly equalized less than a minute later when he skated in and shot the puck just wide of Neuvirth.
The Flyers led 9-5 in shots and held a 1-0 advantage at the end of the first.
--Field Level Media
Philadelphia | Pittsburgh | |
Valtteri Filppula 3 | Points | Jake Guentzel 1 |
Valtteri Filppula 1 | Goals | Jake Guentzel 1 |
Valtteri Filppula 2 | Assists | Derick Brassard 1 |
N/A | Power Play Goals | N/A |
Valtteri Filppula 1 | Short Handed Goals | N/A |
Michal Neuvirth .938 | Save Percentage | Matt Murray .875 |
Michal Neuvirth 30 | Saves | Matt Murray 21 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Philadelphia | 25 | 4 | 0-1 | 5-5 | 19 | 20 |
Pittsburgh | 32 | 2 | 0-5 | 1-1 | 11 | 33 |