Joel Farabee and Sean Couturier each had a goal and an assist, and Derick Brassard and Keith Yandle each had two assists for Philadelphia (2-0-1). Martin Jones made 37 saves in his Flyers debut.
"This was a fast-paced game, fast-paced hockey from both teams," Flyers coach Alain Vigneault said. "Both teams forced one another into turnovers. I thought we were able to capitalize on a few more than them."
Karson Kuhlman, Taylor Hall and Brad Marchand scored for Boston (1-1-0). Jeremy Swayman made 19 saves.
"Obviously accountable for what happened," Swayman said. "Thought the team played great, great comeback in the second. I just want to do my job better, give the team a better chance to win."
Farabee started the play that led to the go-ahead goal when he intercepted a pass in the neutral zone and rushed the puck into the Boston end. He centered it to a cutting Atkinson, who pushed the puck past Swayman while falling to the ice to make it 4-3.
The goal came after Marchand tied it 3-3 with 1:00 remaining in the second.
"The most important shifts are after you score a goal or you get scored on," Atkinson said. "It's never a good thing to get scored on in the last minute, but I think we did a good job of responding and staying positive on the bench."
The line of Farabee (six points; three goals, three assists), Brassard (five points; one goal, four assists) and Atkinson (four points; three goals, one assist) has combined for 15 points (seven goals, eight assists) through the Flyers' first three games. They were put together late during the preseason, in part because Atkinson and Brassard were linemates from 2011-13 with the Columbus Blue Jackets.
"I think it's just simple hockey," said Farabee, who has scored in all three games. "We're not really trying to do too much. When we don't have a play we're getting it in and getting on the forecheck, and I think for us that's creating a lot of turnovers and just using that as offense."
Travis Konecny made it 5-3 at 11:17. It was his third straight game with a goal.
