Sutter scored on a power play late in the first period to make the score 1-1 and added his second late in the second period for a 4-2 Hurricanes lead.
Finland native Jussi Jokinen assisted on the first Sutter goal and added a two-man advantage goal that broke a 2-2 tie midway through the second period.
Minnesota scored just 3:33 into the game when Guillaume Latendresse sent a wrist shot past Carolina goaltender Cam Ward But with 10 seconds to go in the period Sutter tipped in a shot and tied the score.
Anton Babchuk's backhander beat Minnesota goalie Niklas Backstrom 5:58 into the second but Matt Cullen tied things for the Wild. The tie lasted less than 90 seconds before Jokinen's goal.
Brent Burns added a power-play goal for Minnesota with 3 1/2 minutes to play.
The same teams are to play a second game in Helsinki Friday as part of the 2010 NHL Premiere series. The NHL also has early season games scheduled in Sweden and the Czech Republic.
Notable deaths of 2014 [PHOTOS]