WATCH: [All Flyers vs. Ducks highlights]
"[Couturier] did a great job battling," said Simmonds, who has four goals, including a hat trick in a 5-3 win in Philadelphia's season opener at the San Jose Sharks on Wednesday. "He stepped to the middle of the ice and he took both guys with him, so that whole left side was open."
Ducks defenseman Cam Fowler scored shorthanded at 3:40 of the third period to tie the game 2-2.
The Flyers (2-1-0) took a 2-1 lead at 13:09 of the second period on Couturier's first goal. Jakub Voracek drove to the net along the left wing with Ducks defenseman Josh Manson on his hip, pulled up just below the goal line, and backhanded a pass to the top of the crease, where Couturier swept the puck through goalie John Gibson's pads.
"I think the speed that we have in this locker room definitely helps. It gives us that extra little push," said Flyers goalie Brian Elliott, who had 21 saves.
Anaheim (1-0-1) didn't get its first shot on goal in the second period until 13:54, and that stretch included its first full two-minute power play.
The Ducks gave up the first goal for the second straight game.
The Flyers took a 1-0 lead one second after a first-period roughing penalty to Ducks center Antoine Vermette expired when Jordan Weal passed the puck from behind the net to defenseman Ivan Provorov, who beat Gibson at 8:19.
Rookie center Nolan Patrick, the No. 2 pick of the 2017 NHL Draft, got an assist on the goal for his first NHL point.