The game, taking place at the Raymond James Stadium, saw the opening goal come just 11 seconds in, as Brandon Hagel put the hosts in front.
After that, Boston dominated proceedings, scoring five times by the middle of the second. Alex Steeves, Morgan Geekie, and Viktor Arvidsson got goals in the second half of the opening period, before Matthew Poitras and Geekie added to the lead after intermission.
Down by four goals, the Bolts had to respond quickly. They immediately started the comeback, as Oliver Bjorkstrand reduced the deficit two minutes later via a power play.
By the end of the second, it was 5-4 thanks to Darren Raddysh and Nick Paul. Tampa did not stop there though, as NIkita Kucherov levelled the score midway through the third to restore parity.
The final 8:10 of regulation, as well as overtime, resolved nothing, meaning a shootout was required. Only Jake Guentzel managed to score his shot, completing the Lightning's comeback in a 6-5 win.
The win brought Tampa back to top spot in the Eastern Conference, winning their third straight. Boston lost ground in the race to top spot in the conference, sitting six points behind the Bolts.
Earlier in the day, the Carolina Hurricanes beat the visiting Los Angeles Kings 3-2 in overtime. They were made to work for the result despite going 2-0 up.
Jordan Staal's power play marker opened the scoring for the Canes. Then, after a scoreless second period, Alexander Nikishin doubled the hosts' advantage. However, the Kings came alive after that, halving the deficit moments later through Samuel Helenius.
Quinton Byfield then made it 2-2 with 3:11 left, silencing the fans at the Lenovo Center. The disappointment was short-lived though, as Carolina foudn their winner 85 seconds into overtime. Sebastian Aho was the hero, converting his chance to end the contest.
It is a third win in their last four for the Canes, keeping up with the Lightning. The Kings dropped to seventh in the Western Conference with the loss.
The other game on Sunday saw Chris Kreider score twice as the Anaheim Ducks held on for a 4-3 success over the Vegas Golden Knights.