It was the second shutout of the season and eighth in the NHL for Gustavsson, who made 16 saves in the first period and 14 in the third.
Kirill Kaprizov scored for the Wild (14-4-4), who had lost two in a row and are 10-1-3 on the road.
Ukko-Pekka Luukkonen made 28 saves for the Sabres (11-10-1), who had won three straight and were shut out for the first time this season.
Kaprizov scored at 6:55 of the first period. He converted on a one-timer from low in the right face-off circle off a touch pass from Joel Eriksson Ek on a 4-on-1 rush.
Minnesota defenseman Jonas Brodin was a late scratch because of an upper-body injury.
