Ondrej Palat and Mikhail Sergachev scored for the Lightning (36-9-2), who avoided losing consecutive games for the second time this season. They are 19-2-1 in their past 22 games.
It was the third shutout this season for Vasilevskiy, who allowed four goals Sunday in a 5-1 loss at the New York Islanders, and his second in his past five games.
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"You know he's going to bounce back after an effort that isn't acceptable, up to his standards and his mind," Tampa Bay captain Steven Stamkos said of Vasilevskiy. "He always bounces back and finds a way. He was First Star (of the game) for a reason."
Anton Khudobin made 19 saves for the Stars (23-20-4), who have lost three straight games and have scored one goal or fewer in four of their past five.
"We played a good brand of hockey," said Dallas coach Jim Montgomery, who after a 3-1 loss to the St. Louis Blues at home on Saturday said that the organization suffers from a culture of mediocrity. "We're just struggling to score goals right now. We've got to keep with it. And if we keep playing like that, the way we're skating and creating really good looks, we're going to end up on top of a lot of games."
Palat scored with a one-timer from the right face-off circle off a backhand pass on the rush from Stamkos to give the Lightning a 1-0 lead at 7:55 of the first period. Stamkos skated down the left side, drawing attention from the three Stars who were in the zone.