Couturier's 29th goal came when he blasted the puck from the right circle to the far side past goaltender Matt Murray, who also gave up James van Riemsdyk's tying goal with 18.8 seconds left in regulation.
Flyers goalie Carter Hart, in just his second start in 11 games, made 41 saves.
The game was played 16,000 days after Philadelphia (35-29-8, 78 points) last won the Stanley Cup in 1975, and the two points helped keep the Flyers' playoff hopes alive. They are six points out of an Eastern Conference wild-card spot with 10 games to play.
Islanders 3, Wild 2 (OT)
Brock Nelson scored at 1:04 of overtime as New York defeated host Minnesota, which had twice erased one-goal deficits.
Jordan Eberle and Anders Lee also scored, and goaltender Thomas Greiss made 32 saves as the Islanders won for the fifth time in their past seven games and bounced back from a disappointing 2-1 loss to the Red Wings on Saturday in Detroit.
Zach Parise and Jared Spurgeon scored for Minnesota, which lost for the fourth time in its past five games. Goalie Alex Stalock, making just his second start in the past 14 games, stopped 17 of 20 shots.
Golden Knights 6, Oilers 3
Mark Stone, Cody Eakin, Brayden McNabb and Alex Tuch also scored goals, and William Karlsson and Reilly Smith each had two assists for the Knights (40-27-5, 85 points), who won for the eighth time in nine games.
Ryan Nugent-Hopkins, Leon Draisaitl and Kris Russell scored goals, and Connor McDavid had two assists for Edmonton (32-33-7, 71 points), which remained seven points out of the second wild card in the Western Conference. It was the eighth straight multi-point game for McDavid, who now has 105 points this season.
Sabers 4, Blues 3 (SO)
Sam Reinhart scored in the third round of the shootout as host Buffalo snapped a seven-game skid with a victory over St. Louis.
Reinhart nearly won the contest in the waning moments of overtime, but his shot caromed off the left post.
Buffalo captain Jack Eichel and St. Louis forward Tyler Bozak each scored in the first round of the shootout. Reinhart's bid banked off Jake Allen's glove and into the net to set the stage for former Sabres forward Ryan O'Reilly, whose shot deflected off the crossbar to end the game.
Avalanche 3, Devils 0
Tyson Barrie scored two goals, Philipp Grubauer had 22 saves and Colorado kept its postseason hopes alive with a win over New Jersey in Denver.
Nathan MacKinnon had a goal and an assist to give him 91 points on the season. He joins Hall of Famer Peter Forsberg as the only two Avalanche players to have consecutive 90-point seasons. Colorado trails Arizona by four points in the Western Conference wild-card race. The Avalanche host the Coyotes on March 29.
Cory Schneider stopped 31 shots for the Devils, who had their two-game winning streak snapped. New Jersey has not won three straight since late December.
Canucks 3, Stars 2 (SO)
Josh Leivo's goal in the fourth round of the shootout lifted visiting Vancouver past Dallas.
The Canucks posted their second win in three games, while the Stars took their second straight loss, which prevented Dallas from gaining ground on St. Louis for third place in the Central Division.
Tim Schaller led the Canucks with two goals, marking the first two-goal game of his NHL career. Jamie Benn and Taylor Fedun tallied for the Stars.
Ducks 3, Panthers 2
Adam Silfverberg scored with 4:43 to play as Anaheim continued to play spoiler with a victory over visiting Florida.
Devin Shore and Adam Henrique also scored goals, and Ryan Getzlaf had a pair of assists as the Ducks (30-35-9, 69 points), won for the sixth time in their past nine games, with all six of those victories coming against teams either currently in a qualifying position for the playoffs or fighting to become eligible.
The Panthers (32-28-12, 76 points) were the furthest out from eligibility in that group, but their hopes had been raised with a four-game winning streak that had them believing they could grab the second wild-card spot in the Eastern Conference.
--Field Level Media