Kucherov won it on a shot from the left corner that deflected off the skate of Vegas defenseman Nicolas Hague past Hill.
The goal came 1:27 after Brandon Hagel tied the game 3-3 with a shot that beat Hill to the top corner.
J.J. Moser had a goal and two assists, and Andrei Vasilevskiy made 22 saves for the Lightning (3-0-0).
Jack Eichel and Shea Theodore each had two assists for Vegas (3-2-0). Adin Hill made 21 saves.
Hagel tied the game 3-3 at 17:38 of the third period with a shot that beat Hill to the top corner.
Brayden McNabb gave Vegas a 1-0 lead at 5:26 of the first period with a shot from the high slot that beat Vasilevskiy on the glove side.
Moser tied it 1-1 at 10:21 with his first goal of the season, taking the puck over Brett Howden and beating Hill five-hole.
Pavel Dorofeyev put Vegas ahead 2-1 at 15:20 with a power-play goal, a one-timer from the right circle off a pass from Theodore.
Kucherov tied it 2-2 at 19:47 when he finished off a 2-on-1 with Jake Guentzel.
Ivan Barbashev gave the Golden Knights a 3-2 lead with a goal at 1:59 of the second period, a shot from the low slot off a pass from Theodore.
