By the time they got themselves turned around again to make a late charge against Minnesota, they were too late to offset the damage from their trips to the penalty box in the first and second periods and the Wild held on for a 6-4 victory.
Starting their second season, the Wild (3-0-1) won three straight games for the first time in franchise history.
The Hawks (2-3) should be so fortunate.
A night after shutting out Phoenix, Hawks goalie Jocelyn Thibault was taken out after surrendering four goals in 26 minutes 31 seconds.
Minnesota set another team record by scoring three power-play goals in the first period. Sutton started at 11 minutes 24 seconds; Jim Dowd kept the scoring going at 16:59; and Stacy Roest concluded the binge at 18:39.
Twenty seconds after Roest put the puck in the net, Tony Amonte scored his second goal in two nights.
In the second period, Felipe Kuba's even-strength goal at 6:31 put the Hawks down 4-1 and prompted coach Brian Sutter to send in backup goalie Steve Passmore.
Less than six minutes later, Andrew Brunette scored on a power play, but that was countered by Eric Daze's power-play goal later in the period.