Brodziak opened the scoring at 1:10 of the first period and then gave the Wild a 5-1 cushion in the third that held up against two late goals by the Oilers.
Niklas Backstrom made 16 saves for Minnesota. Edmonton's Devan Dubnik tallied 25 saves on 30 shots.
Zach Parise, Charlie Coyle and Mikael Granlund added goals for the Wild, who were tied with San Jose for the No. 5 seed in the Western Conference heading into the Sharks' Tuesday game against Los Angeles. The Sharks and Wild meet Thursday.
Nail Yakupov had a goal and an assist for the Oilers, who lost their sixth in a row. Shawn Horcoff and Magnus Paajarvi scored late third-period goals to prevent a rout.