Already up 2-0 at the start of the second period, Kucherov scored at 0:27 and 10:21 of the middle frame as Tampa Bay's lead ballooned to four. He then completed his hat trick 14 minutes into the third on a one-timed blast from the point that got through traffic in front.
In addition to Kucherov's big night, Steven Stamkos and Alex Killorn each registered three points in the win.
With Tampa Bay dominating possession all night, Andrei Vasilevskiy faced only 18 shots and stopped 17. The only goal he allowed came on a power play from Vinnie Hinostroza five minutes into the third period.
The Lightning will now return home for a two-game homestand against Vancouver and Dallas that begins on Thursday night.
Scoring Summary
1st Period
TBL 1, BUF 0
2:16 Ondrej Palat (15) - Victor Hedman, Jan Rutta
Two Buffalo skaters were drawn to Victor Hedman along the near wall, but Hedman was able to lift a feed through to Ondrej Palat in the slot. Palat settled the feed and quickly snapped the shot home to open the scoring. The goal extends Palat's scoring streak to five games.
TBL 2, BUF 0
10:45 Anthony Cirelli (10) - Mikhail Sergachev, Alex Killorn
From his own defensive zone, Alex Killorn worked the puck through a defenseman's legs to spring Mikhail Sergachev on an odd man rush. Sergachev fed Anthony Cirelli in the slot and he beat Ukko-Pekka Luukkonen to double Tampa Bay's lead just past the midway point of the first period.
2nd Period
TBL 3, BUF 0
0:27 Nikita Kucherov (2) - Steven Stamkos, Brayden Point - PPG
At some point in the first period, Luukkonen suffered a lower-body injury, forcing the Sabres to turn to Malcolm Subban between the pipes to begin the second period. Nikita Kucherov welcomed him to the game with a wicked wrist shot from the top of the right circle in the waning seconds of a power play chance that carried over from the first period.
TBL 4, BUF 0
10:21 Nikita Kucherov (3) - Corey Perry, Steven Stamkos
Fresh out of the box after the Bolts killed off his hooking minor, Kucherov scored his second goal of the period to grow the lead to four. Corey Perry threaded a pass through the slot to Kucherov on the far side of the ice and he put his shot upstairs on Subban.
TBL 5, BUF 0
14:04 Steven Stamkos (17) - Alex Killorn, Anthony Cirelli
The Lightning's second period dominance was on display with Stamkos adding their fifth goal of the night at 14:04. Five gassed Buffalo skaters got caught on the ice for an extended shift with the Bolts pressing hard in the offensive zone. With the Sabres chasing the puck down low, Alex Killorn fed Stamkos unmarked in the slot for a goal that rang off the post and in.
3rd Period
TBL 5, BUF 1
5:31 Vinnie Hinostroza (7) - Colin Miller, Will Butcher - PPG
The Sabres got on the board with a power play goal just over five minutes into the third period. Will Butcher's initial shot was stopped by Andrei Vasilevskiy, but the rebound was kicked directly to an open Vinnie Hinostroza at the far post.
TBL 6, BUF 1
13:57 Nikita Kucherov (4) - Victor Hedman, Alex Killorn
Kucherov completed his hat trick with just over six minutes remaining in the third on a one-timer from the point that got through Subban with Killorn providing the screen in front of the net.