Cullen opened the third round of the tiebreaker by coming in quickly on Nashville's Pekka Rinne, slowing down, stickhandling and putting a shot past the goalie's right pad.
The Predators' David Legwand followed by clanging a shot off the post and Wild claimed their fourth win in six contests.
Backstrom bounced back with a solid game after being pulled from Minnesota's Tuesday loss to Chicago, improving to 22-4 lifetime against the Predators.
Zach Parise also scored for the Wild.
Legwand scored in regulation for the Predators, who lost for the fourth time in five games.
Rinne finished with 28 stops for Nashville.
Notable deaths of 2014 [PHOTOS]