Brink said he was able to take some of the good things he did playing with Cates and Foerster and replicate them with new linemates.
"Those two guys, Cates and Foerster, are great to play with," he said. "But 'Lycksy' and Poehls played great today. Unbelievable plays on the goals and made life easy on me."
Goncalves put the Lightning ahead 1-0 with his first NHL power-play goal at 8:29 of the first period.
Brink tied it with a shot from the right side at 14:20.
Girgensons made it 2-1 when he scored from the slot 15 seconds into the second period.
Brink's second of the game tied it 2-2 at 10:55. He took a rink-wide pass from Sean Couturier, skated in and beat Johansson in close with a backhand.
Poehling put the Flyers ahead 3-2 at 19:01 when Lycksell found him alone in the slot, and the forward deked around Johansson and tucked the puck inside the post.
"I saw he was out of his crease a bit and had a lot of time and space," Poehling said. "I've had that a few times and hasn't worked out. So it's nice that it did that time."
Cooper called the defensive breakdown on the play "inexcusable."
"That should never happen, that goal," he said. "One of them chasing the puck and three of us, and then all of a sudden within five seconds one of them all alone at our net. That's inexcusable. You get these tight games and you get down the stretch, you can't make mistakes like that."
Atkinson tied it 3-3 at 3:50 of the third period with a shot from the inside edge of the right face-off circle. It was the forward's first game in Philadelphia since the Flyers bought out the final season of his contract in June.
"Lot of emotions, family is in town, friends," he said. "Didn't get the job done. ... Have to do a little bit better at shooting when we have opportunities. That's what kicked us a little bit.
"I think we're just trying to pass the puck into the net sometimes instead of capitalizing on our opportunities. It's hard to score goals in this league, so when you have the opportunity to shoot the puck, we have to do more of that and usually good things happen."
NOTES: Brink had his second two-goal NHL game; he also scored twice against the Minnesota Wild on Oct. 26, 2023. ... Philadelphia also ended a five-game home losing streak. ... Tampa Bay defenseman Victor Hedman got his 40th assist of the season on Atkinson's goal. It's his ninth 40-assist season, passing Martin St. Louis and Kucherov for the most in Lightning history. ... Atkinson had been recalled from Syracuse of the American Hockey League on Wednesday. He had 78 points (36 goals, 42 assists) in 143 games with the Flyers.