Fortier's second goal gave the Lightning a 6-5 lead 11:54 into the third period, and he completed the hat trick with a shorthanded goal at 16:33 to make it 7-5.
Perry, who signed a two-year contract July 29, made it 8-5 at 18:18.
Brayden Point and Ondrej Palat each had a goal and two assists, Nikita Kucherov had three assists, and Hugo Alnefelt made 39 saves for Tampa Bay.
C.J. Smith had two goals and an assist, and Antti Raanta made 18 saves for Carolina.
