Joe Pavelski pulled a goal back for Dallas in the second period on a power play before Mattias Janmark added a second in the third period for the Stars.
Tampa Bay goalie Andrei Vasilevskiy finished with 27 saves, including 17 made during a ferocious Dallas onslaught in the second period.
"This was an emotional game," Lightning forward Anthony Cirelli said. "There were ups and some downs. It's always good to come out with the win.
"I thought we did a pretty good job of weathering the storm there. Whenever you can come out with a victory is a positive for us, so we'll take some video, we'll look at some good and the bad things, and move on."
Russian winger Nikita Kucherov was the architect of Tampa Bay's first period scoring blitz, providing two assists as his team scored three times in just 3min 53 seconds.
Point and Palat scored Tampa Bay's opening two goals on power plays.
"We had some good looks in the first game, we just couldn't score," Kucherov said.
"I think we just stuck to what we have to do, keep it simple, shoot the puck at the net, get those rebounds.
"Just keep it simple, that's the key to our success today and fortunately we got two goals today," he said.
Dallas coach Rick Bowness, meanwhile, bemoaned his team's error-strewn first period that allowed Tampa Bay to take control of the contest.
"They won the first, we won the second," Bowness said.
"But you can't be doing what we did in the last half of that first period. It's very simple to explain.
"The penalties got us in trouble, no question. We had a lot of unforced turnovers," he said.
The Lightning are seeking to add a second title to the one they captured in 2004. They fell in six games to Chicago in the championship series in 2015.
This year's Stanley Cup finals are being held at a neutral site with no fans in attendance after the season was halted earlier this year due to the coronavirus pandemic.
Game three in the best-of-seven series takes place on Wednesday.