Nate Schmidt scored twice, and Sam Reinhart and Sam Bennett each had a goal and an assist for the Panthers, who are the No. 3 seed from the Atlantic Division. Sergei Bobrovsky made 20 saves.
"It's playoff hockey and everyone has to bring their best," Bobrovsky said. "They're a good team, too, they've got a really good offense. It's going to be competitive. It's one game. We're happy about the result and we're getting ready for the next one."
Jake Guentzel and Brayden Point scored for the Lightning, who are the No. 2 seed from the Atlantic Division. Andrei Vasilevskiy made 11 saves.
Game 2 of the best-of-7 series will be here on Thursday (6:30 p.m. ET; TBS, TRU, MAX, FDSNSUN, SCRIPPS).
"A series isn't won in one game, that's a positive," Tampa Bay coach Jon Cooper said. "In the end we gave up six goals. We're a pretty decent defensive team and we're good on the PK, and we gave up three on that. We can sit here and dissect this game all we want, bottom line is we lost. We'll turn the page and move on."
Bennett gave Florida a 1-0 lead at 3:44 of the first period when he deflected a pass from Mackie Samoskevich out of the air.