Beck Malenstyn and Josh Doan scored for the Sabres (10-11-4), who had lost two straight, including 5-0 to the New Jersey Devils on Friday. Colten Ellis made 22 saves, and Tage Thompson, Jack Quinn and Noah Ostlund scored in the four-round shootout.
Kirill Kaprizov and Matt Boldy scored, and Filip Gustavssson made 30 saves for the Wild (14-7-5), who extended their point streak to 11 (9-0-2). They were coming off a 3-2 shootout win against the Colorado Avalanche on Friday.
Doan tied it 2-2 at 6:01 of the third period. His wrist shot from the left face-off circle caromed off the glass behind the net and deflected in off Minnesota forward Mats Zuccarello at the right post.
Kaprizov put Minnesota ahead 1-0 at 9:37 of the first period with a one-timer from low in the right circle off a cross-ice pass from Zuccarello.
Malenstyn tied it 1-1 at 12:42, deflecting in the rebound of a Peyton Krebs shot.
Boldy put the Wild back on top 2-1 at 14:05. He stole the puck from Alex Tuch near the right boards on the forecheck, drove to the net and scored through the five-hole.