The Blues were looking to go 5-0-1 in six games under new head Coach Ken Hitchcock, leading 2-1 late in the third period.
But Jason Arnott was whistled for slashing the hand of Minnesota's Pierre-Marc Bouchard with 1:48 left in the game, and Mikko Koivu made St. Louis pay with the equalizer -- his second goal of the game -- 10 seconds later.
After a scoreless overtime, Cullen snapped a shot past Jaroslav Halak in the shootout and Josh Harding stopped Kevin Shattenkirk's last-chance try, handing the Northwest Division-leading Wild their ninth win in 11 games.
Harding stopped 25-of-27 shots to earn the win.
David Backes and T.J. Oshie scored while Halak made 30 saves for the Blues.
Notable deaths of 2014 [PHOTOS]