Tied 1-1 at the start of the second period, the Bolts reeled off four unanswered goals over the final 40 minutes to firmly take control of the game and series.
Kucherov played the role of facilitator, assisting on three Tampa Bay goals, before hitting the empty net late in the third to put the game out of reach.
Vasilevskiy allowed only one goal for the fourth-straight game and finished with 34 saves. He backstopped Tampa Bay's strong finish, stopping all 27 shots he faced in the final two periods.
The Lightning won't have to wait long for their first chance to close out the Cats, as the teams return to AMALIE Arena on Monday night for Game 4.
Scoring Summary
1st Period
TBL 1, FLA 0
13:21 Corey Perry (5) - Ryan McDonagh, Nikita Kucherov
For the third-straight game in the series, Corey Perry lit the lamp first for Tampa Bay. Perry set up in front of the net and his redirection on Ryan McDonagh's point shot got through Sergei Bobrovsky to give the Lightning the lead.
TBL 1, FLA 1
16:07 Sam Reinhart (3) - Jonathan Huberdeau, Aleksander Barkov - PPG
The Panthers ended their power play drought less three minutes later and tied the game at one. Good puck movement from the Cats freed up Sam Reinhart for a one-timer from the slot that Andrei Vasilevskiy couldn't keep out.
2nd Period
TBL 2, FLA 1
2:54 Erik Cernak (1) - Ross Colton, Ondrej Palat
Ondrej Palat's centering feed for Ross Colton ended up being deflected through to Erik Cernak on the left wing. Cernak beautifully picked out the top corner with a wrist shot to put the Bolts back in front early in the second.
TBL 3, FLA 1
10:23 Steven Stamkos (3) - Nikita Kucherov, Victor Hedman
As boos rained down from the crowd, displeased that a potential Florida icing was waived off, the Lightning gathered the puck and quickly headed the opposite direction. Victor Hedman banked an outlet pass off the far wall for Nikita Kucherov at the blue line, and he twisted away from Gustav Forsling to free up space in the middle of the ice. Kucherov then teed up Steven Stamkos for a one-timer from the left circle to double Tampa Bay's lead.
3rd Period
TBL 4, FLA 1
16:06 Nikita Kucherov (4) - Nick Paul - ENG
Kucherov scored the first of two Tampa Bay empty net goals to extend the lead to three with less than four minutes remaining in regulation.
TBL 5, FLA 1
17:51 Steven Stamkos (4) - Nikita Kucherov - ENG
From just inside his own blue line, Stamkos found the empty net for his second goal of the afternoon.
