Carter and Drew Doughty scored regulation goals for Los Angeles, which had just 18 shots. Jonathan Quick made 29 saves for the Kings, including stops against Zach Parise and Mikko Koivu in the shootout.
Trailing 2-1, Carter scored on a rebound of a shot by Matt Frattin at 13:14 of the third period to tie it for Los Angeles.
Cooke scored early in the first period for Minnesota. After a power-play goal by Doughty, Brodin scored with the extra man to give the Wild a 2-1 lead late in the first.