Shortly after Jacob Markstrom stacked his pads to make a save on Miro Heiskanen, Pavelski scored on the rebound of another shot by Heiskanen to tie it 2-2 at 11:41 of the second.
Pavelski then gave Dallas a 3-2 lead at 10:05 of the third period, again getting to a rebound in front on the power play.
"It's another step," Pavelski said. "There was a lot of effort out there tonight from up and down the lineup. Lot of physicality, guys hanging in, taking checks, making plays, giving checks. It was a competitive game. Our group was very competitive, and I like that."
Of the four goals the Stars have scored on Markstrom in the series, Pavelski has three of them. He also scored the game-winning goal in a 2-0 win in Game 2 on Thursday.
"Elite athletes man, they find a way to rise to the occasion," Dallas coach Rick Bowness said. "Joe came up big, but again, it's always around the net. There's a timing to get there. There's a timing anticipating the shot. You can't get there late, you can't get there early, there's a timing with that. Obviously he has it down pat."
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Radek Faksa scored and Heiskanen had two assists for the Stars, who are the first wild card from the West. Jake Oettinger made 39 saves.
Trevor Lewis and Elias Lindholm scored for the Flames, the No. 1 seed from the Pacific Division. Markstrom made 28 saves.
The Stars lead the best-of-7 series 2-1. Game 4 will be here Monday.