Boston won the seven-game series en route to its first Stanley Cup title since 1972.
After a scoreless first period, Boston started the scoring early in the second when Peverley collected a rebound of Marchand's shot that bounced off the post. He put it in the back of the net just 2:04 into the period.
But Tampa Bay answered just a little more than a minute later when St. Louis took a Ryan Malone pass alone down the center of the ice, then beat Thomas high to the stick side, tying the game at one.
Daniel Paille broke the tie just a minute and a half later when he deflected an Adam McQuaid shot from the point past Garon.
It remained a 2-1 game until the third period, when Milan Lucic's wrister from the bottom of the left circle rebounded high into the slot. David Krejci gathered the puck and shot it into the net for a two-goal lead 3:07 into the period.
Peverley put the game away with his second goal of the game at the 8:43 mark of the third.
Tampa Bay didn't get its first shot on net until just over 10 minutes left in the first period...The Lightning are starting the season with a five-game road trip, and won't play at home until hosting Florida on October 17...The game was defenseman Zdeno Chara's 400th with Boston...Tampa Bay went 0-for-5 on the power play and Boston failed to convert on its six power play opportunities.