It was Stamkos' eighth career hat trick, tying him with Martin St. Louis for the franchise record. He had 12 shots on goal.
His third goal came with 1:46 remaining in the second period when he scored a power-play goal with a wrist shot from outside the right circle to give the Lightning a 4-1 lead.
"Overall I thought that this was our best performance of the season so far from start to finish," Stamkos said. "It was our most consistent on offense and defense against a good team. Everyone did their job tonight and we were rewarded with pucks going in the net."
Last season, Stamkos didn't score a goal before registering a hat trick against the Florida Panthers in the fourth game of the season. This season, the Lightning captain failed to score goals in his first two games.
"You definitely put some pressure on yourself to get started early in the season," Stamkos said. "Everyone's probably pushing a little but things happen as long as you are working hard."
Stamkos had a chance for a fourth goal but failed to get a clean shot off on a 3-on-1 against Canadiens backup goalie Dustin Tokarski.
The Lightning turned the game into a rout with third-period goals from Ryan Callahan, Ondrej Palat and Vladislav Namestnikov, who scored his first NHL goal. It was a nice response after a 3-2 shootout loss to the Ottawa Senators on Saturday.
"That was good to see because we had a little lapse in the third period in our last game," Stamkos said. "It was good to finish strong and see some other guys get points on the board."
Callahan's goal came on a superb effort; he skated around Canadiens defenseman P.K. Subban and slipped the puck past Tokarski.
"When we are playing well we were a lot better than what we were tonight," Subban said. "So for us we tell Steven to enjoy his night because the next time it's not going to happen."
It was the first game between the Canadiens and Lightning since Montreal swept Tampa Bay in the Eastern Conference First Round of the Stanley Cup Playoffs last spring. And even though members of the Lightning said it wasn't a revenge game, the scoreboard said differently.
"This was a must-win game for us," Lightning coach Jon Cooper said. "They have been on the road since the start of the season and they were probably a bit tired. When in those situations you have to get those points and we did that. They are a really good team over there."
The Lightning took a 1-0 lead 5:24 into the game on Victor Hedman's third goal of the season when he used a screen from Namestnikov to snap a wrist shot past Canadiens goalie Carey Price, who was pulled after the second period.
Stamkos gave the Lightning a 2-0 lead midway through the first with a shot from close range after Callahan applied pressure in front the net.
"He's getting better with each game," Cooper said. "Stammer is a scorer and in his first two games I think he had seven shots combined. Tonight he had 12 shots and he could have probably had 18 shots. He's definitely trending upward."
"You don't want games like this to happen," said Max Pacioretty. "We show what we are capable of doing when we play the right way. But it was a huge letdown tonight. We just have to control our emotions, never get too high or low."
Hedman had three assists to give the defenseman seven points through three games to lead the NHL. It was the first four-point game of his career. He made a beautiful pass to set Stamkos up with a breakaway goal in the second period.
"I think this was the best game I've played so far this short season," Hedman said. "I thought we played to our strengths tonight, we possessed the puck and drew some penalties and capitalized on them. We're looking forward to [the game Tuesday against the New Jersey Devils]."
Price had 19 saves before being pulled; Tokarski stopped 15 of 18 shots in the third period.