Wild defenseman Brent Burns partially fanned on a clearing attempt from behind his own net, putting the puck right on the stick of Vermette. The Blue Jackets' center fired it past Niklas Backstrom for the game-winner.
The overtime goal salvaged only the second win in 11 games for Columbus, which had squandered a two-goal lead late the contest.
Rick Nash scored two goals and added a pair of assists, Sami Lepisto contributed three helpers and Mathieu Garon stopped 32-of-36 shots for the Blue Jackets.
Antti Miettinen and Pierre-Marc Bouchard each scored in the final 5:07 of regulation to rally the Wild, but they couldn't finish and fell to 2-6-2 in their last 10 games.
Backstrom ended with 28 saves for the Minnesota.