Casey Mittelstadt and Alexander Nylander each scored his first NHL goal for Buffalo (25-44-12), which is 2-8-0 in their past 10 games and will finish last in the NHL.
"We knew what to expect, obviously they are a team that can score a lot," Sabres center Jack Eichel said. "We put ourselves behind the eight ball a little bit and probably gave them too much."
Anthony Cirelli tied the game 5-5 at 6:15 of the third period off a tip from a shot by Alex Killorn. Brayden Point made it 6-5 at 7:02 with a one-timer that deflected off the shoulder of Sabres defenseman Brendan Guhle. Victor Hedman scored an empty-net goal at 18:34 for the 7-5 final.
It was the Sabres' first regulation loss when leading to start the third period (16-1-5).
"We just kind of regrouped [in the third period]," Cirelli said. "We know we're a good team. We know the way we need to play. We went out there in the third and lucky for us we got a couple of goals and got the win."
Cirelli scored 30 seconds into the first period to give the Lightning a 1-0 lead. Mittelstadt, in his fifth NHL game, scored on a power play at 2:57 to tie it 1-1.
Kucherov made it 2-1 with an unassisted goal at 4:35, before Nylander, in his sixth NHL game, tied it 2-2 at 6:32. Yanni Gourde set a Lightning rookie record with his 25th goal at 9:40 for a 3-2 lead; it was his first in 14 games.
"I hadn't scored in a long time so it felt very good to finally score," Gourde said. "I was more happy to score a goal than to break the record. It was such a long stretch to go without a goal and I was happy to get one. But obviously my teammates played amazing tonight. Killorn and Cirelli played very well. It was good for the group to get us going."
Dan Girardi scored off on a one-timer at 1:04 of the second period to give Tampa Bay a 4-2 lead. Jason Pominville made it 4-3 at 8:59, Jordan Nolan tied it 4-4 at 15:58, and Kyle Okposo scored a power-play goal at 18:45 to make it 5-4.
"We had a [penalty] kill and that was huge for us," Mittelstadt said. "It carried over and obviously we got a few [goals] there. We've just got to do a better job in the third."
Andrei Vasilevskiy made 27 saves for his NHL-leading 44th win. Gourde had two assists to reach 64 points, also a Lightning rookie record.