Couturier sealed the hat trick with a goal in the third period.
The Penguins play the victor of the Washington-Columbus series in the Eastern Conference playoffs.
Couturier revealed to reporters following the game that he had played Games 5 and 6 with a torn MCL, an injury he said would have taken four weeks to heal if suffered during the regular season. Washington leads that series 3-2.
Trailing 5-4, the Flyers failed to connect on a four-on-three power play midway through the third period.
That was in stark contrast to Game 3, when the Flyers had four really good chances on Pittsburgh goalie Matt Murray in the opening 14 minutes, only to fall behind 2-0 on a Phil Kessel goal shortly thereafter.
In a memorable (but for many, also forgettable) Game 6 in Philadelphia, the Penguins proved too opportunistic - and talented - for the fledgling Flyers to defeat to force a Game 7, faltering 8-5 to end their up and down season.
Do the Philadelphia Flyers have a shot to take the series to a decisive Game 7?
Couturier was the hero for the Flyers in Game 5 following the freak accident in practice led to a knee injury that forced him to miss Game 4. It tied the franchise record for goals against in any length of series, joining the 1979 New York Rangers team that scored 28 in a five-game series win.
"Mur-ray... Mur-ray... Mur-ray", the fans chanted sarcastically. And 47 seconds after Crosby's equalizer, Carl Hagelin put the Penguins back where they are used to being in South Philly - ahead. Olli Maatta's shot from well above the left circle hit the left post and caromed off Michal Neuvirth and into the net to make it 4-4. "It happened quick. It looked like we were in control of the game and they got two quick goals there".
Couturier was everywhere. Winning puck battles. They have combined for 18 points in 5 games and share a combined plus-11 rating. Couturier scored his second of the game on a attractive breakaway to open the second period and Scott Laughton scored on a long wrister and almost pulled off a Lambeau Leap over the boards in celebration.
Wayne Simmonds created a turnover that led to Couturier's goal.
But the Penguins, who were missing star center Evgeni Malkin because of an unspecified injury, didn't get rattled.
Pittsburgh Hockey Now asked Sidney Crosby about the anomaly. Filppula failed to pick up Crosby, who was all alone. A Claude Giroux turnover started the sequence, which gave the Penguins a 2-1 lead. Defensemen Robert Hagg, Travis Sanheim and winger Oskar Lindblom have also made an impression on the Flyers, with more promising guys in the pipeline.