Chicago had opened the lockout-reduced season with a 6-0-0 record to tie San Jose for the best record in the NHL, but Minnesota ended the run thanks largely to a 28-save relief appearance from goalie Niklas Backstrom.
The Finn was forced into action after Josh Harding allowed two goals on just four shots.
"I've never done that, pulled a goalie after two goals," Wild coach Mike Yeo said. "It's not a decision I'm sitting here feeling good about. I just felt it's what we needed."
Cullen scored the opening goal early in the first period before Andrew Shaw and Jonathan Toews replied for the visitors to give the Blackhawks a 2-1 lead and chase Harding from the game.
Minnesota's Cal Clutterbuck equalised 59 seconds into the second period to usher in 44 minutes of scoreless action that passed through overtime and into the shootout.
After each team scored once through two rounds, Cullen slipped his shot past Corey Crawford before Patrick Sharp hit the crossbar to end the contest.
"Those are the tight games the last three or four games that we've found a way to win," Toews said. "We were close again. We had a chance in overtime and in the shootout, we just came up a little bit short.
Chicago out-shot Minnesota by a combined 17-6 in the third period and overtime but could not make a breakthrough.
The Wild (4-2-1) have now won two straight since dropping three in a row and are 4-1-0 at home.