Palat gave the Lightning a 1-0 lead at 7:29 of the first period off a one-timer from the right face-off circle off a pass by Victor Hedman 10 seconds into a power play.
Palat, who has replaced injured Nikita Kucherov on the first power-play unit, leads the Lightning with an NHL career-high six power-play goals. He scored five in 56 games in 2016-17.
"It's always nice when you score on the first PP," Palat said. "We had a lot of chances on the power play, especially in the second and third period, so we could be better there.
"In the offseason I didn't know I was going to play [in Kucherov's] spot. But I'm working on [the one-timer] in practice and knowing just when to shoot the puck."
Gourde scored into an empty net with five seconds remaining in the third period for the 2-0 final.
Dallas was 0-for-3 on the power play and is 0-for-9 against Tampa Bay this season. The Lightning were 1-for-5.
"We had some great looks tonight. A couple of times we missed the net and a couple of times he makes a good save," Stars forward Joe Pavelski said. "It's one of those moments where it's not going in right now. We're giving up 1-2 goals ... we've got to win these games."
Tampa Bay is 16-for-16 on the penalty kill in the past four games.
"The more you play with somebody, the more comfortable you are and understand where they are going to be in certain situations," Lightning forward Blake Coleman said of the penalty kill. "While the forwards are important, it's obviously a unit. [The defensemen have] been great, goaltending has been great. You can't have a great kill without all five guys doing their part."
NOTES: Vasilevskiy is 2:01 from John Grahame's Lightning record shutout streak of 202:46 set over four games from Jan. 17-29, 2006. … Vasilevskiy has not allowed a goal in 10 periods, dating to the second period of a 4-2 win against the Hurricanes on Feb. 22. He has had five consecutive shutouts in the NHL, the latest was Jan. 9-11, 2020. ... Tampa Bay has outscored its opponents 17-3 during its winning streak.