Jared Spurgeon tied it 3-3 at 17:58 of the third period, backhanding in a rebound of an Eriksson Ek shot.
The Wild killed off a penalty for the final 1:56 of overtime, then Kirill Kaprizov scored the only goal of the shootout.
Yakov Trenin also scored for the Wild (30-14-10), who are 4-1-1 in their past six games. Jesper Wallstedt made 29 saves.
Teuvo Teravainen, Ryan Donato and Ilya Mikheyev scored for the Blackhawks (21-23-9) who have lost three straight (0-1-2). Spencer Knight made 20 saves.
Teravainen gave Chicago a 1-0 lead at 1:19 of the first period after Bedard had a tip with Teravainen crashing the net for the backhanded rebound. Teravainen had missed the previous six games with an upper-body injury.
Donato extended the lead to 2-0 at 15:14 after a Brock Faber turnover found Donato at the left hashmarks for a slap shot.
Mikheyev took a pass from Alex Vlasic at the offensive blue line and sniped one cleanly past Wallstedt at 5:54 of the second period for a 3-0 lead.
Trenin cut the deficit to 3-1 at 12:33, coming off the half-wall for a wrist shot stick side.
Eriksson Ek brought it to 3-2 at 3:57 of the third period. Quinn Hughes took a shot from the point that went off Marcus Johansson’s skate right to Eriksson Ek at the doorstep.
