Claude Giroux also had a hat trick in a wild game that ended with Philadelphia winning on Pittsburgh's ice for the seventh time in eight tries.
Philadelphia has come from behind in both games of the opening-round playoff series to take a 2-0 lead. The Flyers have a chance to close out the series at home when they host the third and fourth games on Sunday and Wednesday.
Philadelphia overcame a 3-0 deficit to win the first game of the series and the Flyers found themselves behind 2-0 midway through the first period Friday.
Pittsburgh still had a 3-1 lead through the opening 20 minutes, but Giroux scored on the power play and shorthanded as well to get the game tied midway through the second.
Chris Kunitz put the Penguins back in front, after which Couturier deadlocked the game with just 3 seconds left in the middle frame.
Pittsburgh again took the lead early in the third on Tyler Kennedy's goal, but it was all Philadelphia after that.
Couturier tied it once more just 14 seconds after Kennedy had scored and Jaromir Jagr gave the Flyers their first lead of the game with 10:47 to play.
Another goal by Couturier with 1:49 remaining and an empty-net tally from Giroux completed the scoring. The three goals from Couturier were the first playoff scores of his career.
Both goaltenders -- Ilya Bryzgalov for Philadelphia and Marc-Andre Fleury for Pittsburgh -- made 23 saves.