The Dallas penalty kill went 15-for-15 in six games in the first round against the Nashville Predators and denied St. Louis on its first opportunity.
Tarasenko scored with a power move from the right side around defenseman Miro Heiskanen to make it 3-1 at 3:51 of the third. Tarasenko had another cutting power move in the second he didn't score on.
"I obviously scored (in the third)," Tarasenko said. "Cut the middle. It's nice, but guys know you do this a lot, so you just try to take the puck wide, take it to the net, use the body, and good thing it works at the right time."
It was Tarasenko's seventh multigoal game in the NHL playoffs, first since April 28, 2017, against the Predators.
The Stars nearly tied it at the final horn, but Benn fanned on a puck thrown into the crease.
"We stuck with it," Benn said. "I thought our third was not bad. We gave up that one early, but it's going to happen. ... They're a good team. Tarasenko made a great move, and you've got to give credit to them."