Game 6 of the best-of-7 series is in Edmonton, the hub city for the Cup Final, on Monday (8 p.m. ET; NBC, CBC, SN, TVAS).
Sergachev scored on a one-timer from the left point at 3:38 of the third period to give the Lightning a 2-1 lead.
"I mean, we were close, 10 minutes away," Sergachev said. "But the job's not finished. They scored a goal. Didn't hold the lead."
Perry gave the Stars a 1-0 lead at 17:52 of the first, scoring off a loose puck in the slot. He has scored three goals in his past two games after not scoring in the previous 19.
"You just have an understanding that whatever it takes, he'll try to find a way to win," Pavelski said. "That's what he does, that's the way you can feel the mindset in the locker room, on the ice. Whatever it takes. It's been there, and no bigger moment for us than tonight. Showed up and got it done."