The Wild won their second in a row and moved two points in front of Chicago for the best record in the Western Conference.
Brodziak tied the game at 2-2 with 5:09 left in the second period after Minnesota had fallen behind by two goals early in the middle stanza. Devin Setoguchi began the Minnesota comeback with 8:29 to go in the second.
Niklas Backstrom made 28 saves in regulation and overtime before giving up three goals in the shootout.
Khabibulin stopped 27 shots in Edmonton's third straight loss.
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