The Red Wings equaled their entire goal output from their first three games, two of which they lost.
Brunner scored just 13 seconds into the middle period after neither team could produce a goal in the first.
Todd Bertuzzi gave Detroit a 2-0 lead 42 seconds later and although the Wild came back to tie it, Henrik Zetterberg and Pavel Datsyuk came through late in the second period to put the Red Wings in front 4-2.
Bertuzzi added a second goal 4:18 into the third. Jimmy Howard made 23 saves in collecting the victory.