Kucherov pushed the lead to 3-0 on the power play 27 seconds into the second period, scoring from the top of the right circle after an easy zone entry. He extended it to 4-0 at 10:21 moments after leaving the penalty box, tapping in a cross-ice pass from Corey Perry at the left of the net.
Killorn sent a pass from behind the net to Stamkos coming into the slot to make it 5-0 at 14:04.
Hinostroza capitalized on a rebound that bounced to him at the bottom of the left circle on the power play to cut it to 5-1 at 5:31 of the third period.
Kucherov scored from the high slot at 13:57 for the 6-1 final.
"It was our own fault," Sabres forward Zemgus Girgensons said. "We gave some high-danger scoring chances to one of the top scoring teams. It was our own fault. At this point it was just mental to get back into it. It was a tough night. That's not how we want to play."
Palat left the game in the second period with an undisclosed injury.
"'Pally's' working through something," Cooper said. "So we just thought that anytime we can give him some rest we'll give him some rest, and that's what we did tonight."
NOTES: Kucherov tied Tyler Johnson for the fourth most hat tricks in Lightning history behind Stamkos (9), Martin St. Louis (8) and Vincent Lecavalier (6). … Vasilevskiy has won 20 or more games in six straight seasons. … Lightning defenseman Zach Bogosian was plus-2 with two hits in 18:25 in his first game against the Sabres since being placed on unconditional waivers Feb. 21, 2020. … Palat played his 589th game to tie Johnson for seventh place in Tampa Bay history. … Forward Jack Quinn, selected by the Sabres with the No. 8 pick in the 2020 NHL Draft, played 16:30 and had three shots on goal in his NHL debut.