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.