Brandon Hagel had two goals and an assist, Jake Guentzel added four assists, and Andrei Vasilevskiy made 28 saves for the Lightning (12-4-2), who are 5-1-2 in their past eight games.
Josh Doan, Jason Zucker and Tage Thompson scored for the Sabres (8-7-3), who have lost three of four. Ukko-Pekka Luukkonen allowed five goals on 14 shots before he was replaced by Colten Ellis (three goals on eight shots) early in the third.
Scott Sabourin gave Tampa Bay a 1-0 lead at 18:49 of the first period. After a rebound rolled out to the slot, he quickly snapped it glove side.
Carlson scored on the power play 49 seconds later to make it 2-0 at 19:38. He received a cross-crease pass from Guentzel at the back door, settled it and lifted it into the net.
Hagel pushed the lead to 3-0 from the slot at 4:33 of the second period when he snapped a loose puck past Luukkonen’s glove.
Yanni Gourde took a pass from Guentzel on an odd-man rush and beat Luukkonen stick side to increase it to 4-0 at 13:02.
That prompted Buffalo coach Lindy Ruff to call his timeout.
Doan then cut it to 4-1 on the power play, redirecting Ryan McLeod’s pass from the left circle by Vasilevskiy’s left skate at 15:54.
Zucker narrowed it to 4-2 at 19:32, finishing a backhand feed from Noah Ostlund off the rush from the left circle.
Thompson’s one-timer from the top of the right circle went off the far post and in 33 seconds into the third period to pull the Sabres within 4-3.
Kucherov made it 5-3 at 1:25 on a breakaway, beating Luukkonen with a backhand under his left arm and ending the goalie’s night.
Kucherov then one-timed a pass by Carlson from the right circle to extend it to 6-3 at 4:11 and completed the hat trick from the same spot on the power play, off another Carlson feed, to widen the margin to 7-3 at 14:13.
Hagel scored from the slot at 15:06 off a Kucherov drop pass to make it an 8-3 final.
