Marc-Andre Fleury made 29 saves, and Joel Eriksson Ek and Jared Spurgeon scored for Minnesota (27-17-4), which had lost three of four.
It was the last game before the All-Star break for the Wild. They next play at the Arizona Coyotes on Feb. 6.
"I'm just glad we didn't get away from our game," Wild coach Dean Evason said. "It was close, it was tight. We made some mistakes for sure, but guys were backing each other up and picking each other up on the bench. We didn't have the jump that I think we normally have, so we grinded, which is how we're going to win hockey games anyway. They put a little more emphasis on that, and we got it done."
Jack Quinn and Zemgus Girgensons scored, and Ukko-Pekka Luukkonen made 33 saves for Buffalo (26-19-4), which extended its point streak to seven (5-0-2).
"We generated enough to win," Sabres coach Don Granato said. "At least we got a point out of it, which is big. The road trip, we had seven out of eight points, so it's a good trip. These guys should walk out with their heads up."