Brayden Point stole the puck from Shea Theodore to initiate a 2-on-1 rush, and Kucherov chipped the puck past Golden Knights goalie Carl Lindbom on a return pass from Brandon Hagel.
Kucherov, who reached 1,000 NHL points the previous night in a 4-3 win against the Anaheim Ducks, had what appeared to be the go-ahead goal taken off the board at 13:38 of the third period when Vegas successfully challenged for offside.
Hagel also scored for the Lightning (3-4-2), who have won two straight for the first time this season. Andrei Vasilevskiy made 18 saves in his first win of the season.
William Karlsson scored for Vegas (5-1-3), coming off its first regulation loss of the season on Saturday at the Florida Panthers. Lindbom made 26 saves in his NHL debut.
Hagel scored his first goal of the season to give the Lightning a 1-0 lead at 4:53 of the first period. He scored on a follow up from in front of the net off an initial shot from Jake Guentzel.
Karlsson tied it 1-1 at 18:25 when he whipped a wrist shot from the left circle past Vasilevskiy on the short side.