Anton Babchuk and Jussi Jokinen also scored for the Hurricanes.
Brent Burns scored for Minnesota with 3:21 left to leave the Wild down by one, but Hurricanes goalie Cam Ward made a late save to preserve their win.
Minnesota scored through forward Guillaume Latendresse at 3:33, but Sutter equalized on a power play with 10 seconds left of the first period.
Babchuk put Carolina in front at 5:58 of the second period, only for Minnesota center Matt Cullen to level at 9:52.
Finnish center Jokinen restored the Hurricanes’ lead on a two-man advantage at 11:16, and Sutter made it 4-2 at 18:03.