Marcus Johansson scored, and Marc-Andre Fleury made 27 saves for the Wild (44-24-10), who have lost three in a row (0-2-1).
Minnesota fell two points behind the Colorado Avalanche and Dallas Stars for first place in the Central Division.
"We didn't like the way we started the hockey game, for sure," Wild coach Dean Evason said. "We knew they were going to be desperate, which they were. They came out and played the right way, we did not. By the time we caught ourselves, it was too late. We played well for two periods, but the game is 60 minutes."
Letang put Pittsburgh ahead 1-0 at 15:02 of the first period. He drove around Marcus Foligno to the outer edge of the right face-off circle and scored on a snap shot just inside the right post from a sharp angle.
"I think the start helped," Letang said. "When you come out of the gate and you're on your toes, and you're playing well like this, it sets up success for the rest of the game."