Pierre-Edouard Bellemare scored the go-ahead goal in the third period, and Andrei Vasilevskiy made 33 saves for the Lightning, the two-time defending Stanley Cup champions and the No. 3 seed in the Atlantic Division.
"It's fun to be able to help the team offensively at times," Bellemare said. "Our line, last series we had to kind of manage when we were on the ice, and tonight it was more about trying to build momentum for the team when we're out there. Corey got a big goal, and my goal was just a simple play. Credit to the coach for putting us in an offensive face-off."
Anthony Duclair scored, and Sergei Bobrovsky made 32 saves for the Panthers, who won the Presidents' Trophy with the best regular-season record in the NHL (58-18-6) and are the No. 1 seed in the Atlantic.
Game 2 of the best-of-7 series is here Thursday.
Tampa Bay was 3-for-6 on the power play despite forward Brayden Point being out with a lower-body injury sustained in Game 7 of the first round against the Toronto Maple Leafs. Florida was 0-for-3 and is 0-for-21 in seven Stanley Cup Playoff games.
"I was happy with our game," Panthers coach Andrew Brunette said. "It was the little things. And that's what they do so well, that's why they have won a couple of Cups. It's the little things. It's a little face-off, it's shorthanded trying to score out of nothing. It's special teams. It is all these little things that they excel at and we're still learning. Again, unfortunately, we have to learn again tonight. We'll be better next game."