The Lightning struck first with Nikita Kucherov's goal 13 seconds into the contest, but Buffalo was able to respond at the 3:40 mark when Sam Reinhart capitalized on a rebound to tie the game.
Buffalo was then able to take the lead with an early second-period goal.
A night after being denied on a breakaway chance in Florida, Vesey capitalized on Monday in Tampa. At the 1:48 mark of the period, he beat Vasilevskiy for his first goal as a Sabre.
Buffalo's lead didn't last long, however. The Lightning scored twice in a span of 2:06 to take a 3-2 lead.
First, Mikhail Sergachev put one home from the point on the power play at the 7:20 mark. Then 9:26 into the period, Cedric Paquette's long-range shot through traffic found its way past Hutton .
Victor Olofsson hit the post on the power play with under two minutes to play in the second, for one of Buffalo's best chances.
With 11:43 to play, got loose and Paquette got loose shorthanded, cut in all alone and scored to make it 4-2.
On the same penalty kill, Ondrej Palat scored on a 2-on-1 to give Tampa a three-goal lead with 10:34 remaining.