Dallas is two points behind the Nashville Predators for fourth place, the last remaining Stanley Cup Playoff berth in the eight-team division. The teams play at Nashville on Saturday.
"We've got to find ways to get points," Stars forward Blake Comeau said. "… If you want to make it in the playoffs, you have to go through some really good teams to get where we want to be at. That's winning the Stanley Cup."
Ondrej Palat, Blake Coleman and Brayden Point scored for the Lightning (34-14-2), who have won four straight and have clinched a playoff berth. Tampa Bay is one point behind the Florida Panthers for second place in the Central, three points behind the Carolina Hurricanes for first.
Vasilevskiy leads the NHL with 30 wins and has five shutouts, three against the Stars.
"It was good to see [Vasilevskiy] get in his rhythm," Lightning coach Jon Cooper said. "You could see he was tracking pucks and he was on it. And otherwise there were some big blocks too. Guys were sacrificing tonight, and it was good to see."
Jake Oettinger made 25 saves for the Stars (21-17-12), who have lost two straight and three of five.
"Jake was really good, he did his job," Dallas coach Rick Bowness said. "He gave us an opportunity to get a point out of here. Can't fault him on the goals that went in at all. He made some big saves when he had to. I really liked Jake's game."
Palat gave the Lightning a 1-0 lead at 14:41 of the first period after he forced Stars defenseman Jamie Oleksiak into a turnover and scored from the high slot.
"A big part of our identity is our forecheck, trying to be on top of teams," Cooper said. "When you're on somebody you can try and force them to make mistakes. And that turned out to be a pivotal point in the game because that goal stood up for quite a while."