Paul pushed the lead to 3-1 at 13:37 of the second period when he scored off the rebound of a Charle-Edouard D'Astous shot.
"Obviously the third goal hurt us," Philadelphia coach Rick Tocchet said. "It's one of those games. When you play the game long enough, you're going to have those type of games."
The Lightning scored four times on eight shots in the third period.
"We got a little bit of a lead in the third early and they had to kind of press, and I think we just kind of took advantage of them trying to press," Point said.
Goncalves took a pass from Kucherov on the rush and went backhand-forehand before he slid the puck under the stick of Flyers defenseman Travis Sanheim to score from in front and put Tampa Bay up 4-1 at 2:03.
"Probably early on in my career I don't make that play," Goncalves said. "But being with [Kucherov and Point] ... they're telling me to hold on to it and make plays. When you get a chance to beat a guy, if that's the play, then beat him, that kind of stuff. 'Pointer' has been really great, so has 'Kuch.' Them having belief in me makes my confidence go even more."
Hagel skated through the middle of the ice and finished a pass from Jake Guentzel to make it 5-1 at 3:37.
Tippett got the Flyers to within 5-2 with a power-play goal at 4:45.
Gourde finished a breakaway to make it 6-2 at 9:00 after Philadelphia forward Trevor Zegras turned the puck over at the Tampa Bay blue line.
Goncalves scored his second at 11:00 with a shot from the right circle for the 7-2 final.
NOTES: Kucherov has 36 four-point games, third among active NHL players, behind Connor McDavid (45) and Sidney Crosby (43). He also has 19 points (nine goals, 10 assists) during an eight-game point streak against the Flyers. ... Guentzel had an assist to give him six points (one goal, five assists) during a four-game point streak. ... Hagel has eight points (three goals, five assists) during a five-game point streak. ... Lightning defenseman Darren Raddysh had an assist to extend his point streak to five games (10 points; four goals, six assists). ... It was Cooper's 599th win as an NHL coach, leaving him one shy of becoming the 25th with 600 victories. ... Tampa Bay defenseman J.J. Moser played his 300th NHL game. ... The Flyers played without forward Travis Konecny, who sustained an upper-body injury against the Toronto Maple Leafs on Thursday. He is day to day.