Brunette knocked in a rebound of his own shot against Anaheim's Jean-Sebastien Giguere at 3:08 of the extra period to keep the Wild unbeaten in eight home openers at Xcel Energy Center.
The Wild had the man advantage after James Wisniewski was sent off for roughing.
Niklas Backstrom made 16 saves for Minnesota and Giguere had 28 in his first start of the season.
Minnesota scored three times in the third period to force overtime after the Ducks notched all three of their goals in the second period.
Mikko Koivu, Petr Sykora and Eric Belanger scored for the Wild. Anaheim's goals came from Saku Koivu, Joffrey Lupul and Evgeny Artyukhin.
Celebrity Families of 2014 [PHOTOS]