Chris Driedger allowed five goals on 22 shots for Florida, the No. 2 seed in the Discover Central Division, before being replaced by Sergei Bobrovsky to start the third period. Bobrovsky saved all nine shots he faced, and Jonathan Huberdeau had three assists.
"After the second period a couple guys stepped up," Bobrovsky said. "This team has tremendous ability to come back, it doesn't matter how much of a deficit we have. It's one of those games we get together, one at a time, and we pulled out a W. It's a big win for us."
Hedman had three assists, and Vasilevskiy made 41 saves for Tampa Bay, the No. 3 seed and defending Stanley Cup champion.
Panthers defenseman Keith Yandle was a healthy scratch, missing his first game since March 22, 2009, with the Phoenix Coyotes. His streak of 979 regular-season and playoff games ended two behind Arizona Coyotes forward Phil Kessel for the longest in NHL history.
Sam Bennett gave the Panthers a 1-0 lead at 4:31 of the first period, and Gudas extended the lead to 2-0 at 7:05.
Anthony Cirelli cut it to 2-1 at 1:57 of the second period on the rebound of a shot by Hedman. Ross Colton tied it 2-2 at 5:46 with a wrist shot from the right face-off circle for his first NHL playoff goal.
Steven Stamkos put the Lightning ahead 3-2 with a chip-in off a pass from Ryan McDonagh at 8:38.
Alex Wennberg tied it 3-3 at 12:34 with a power-play goal from the top of the right circle with Owen Tippett setting a screen.
"We were in a good spot after the first period, 2-0, and we do that in the second period," Panthers forward Jonathan Huberdeau said. "We just have to be smarter with decisions, play our game. We had to assess that after the second, not panic, but you don't want to go down 3-0 in the series, and the guys answered the bell in the third period, worked hard and it's a huge comeback for us."
The Lightning regained the lead on Brayden Point's power-play goal at 14:17 to make it 4-3, and Alex Killorn scored on the power play at 18:17 to push it to 5-3.
"It's a series," Killorn said. "Luckily we've been in a lot of these series before. We knew they were going to give us their best effort. They played well tonight, but like I said, it's a series and there's plenty of hockey to be played."