Rick Nash broke out of a slump with two goals and two assists for Columbus, his first goals since Feb. 22, a span of 11 games that matched his career-high drought. Kris Russell and Jan Hejda also scored for Columbus and Mathieu Garon had 32 saves.
Minnesota's Brent Burns turned the puck over in front of his own net after a face-off and Vermette capitalized with the game-winner.
The Wild rallied from a 4-2 deficit with five minutes left in the third period on goals from Antti Miettinen and Pierre-Marc Bouchard.
Brad Staubitz and Cal Clutterbuck also scored for the Wild, who have now lost five straight for the second time this season.
Both teams came into the game with their playoff hopes all but dashed after a stretch of recent poor play caused mainly by a lack of offense.
Columbus managed only 22 goals in its previous 10 games, but appeared to get a lift near the end of the second period.
Trailing 2-1, Blue Jackets winger Derek Dorssett was given a game misconduct after trying to fight Eric Nystrom on his way to the penalty box. Dorssett had just finished a fight with Staubitz.
The game's intensity picked up from there and Russell made it 2-2 four minutes later.
Hejda's slap shot tricked through Nikklas Backstrom's legs early in the third and Nash scored nine minutes later to make it 4-2.
The Wild had managed just four goals in their last four games, but created several chances on Saturday during its late rally.
Staubitz's goal was his first in 80 games and Clutterbuck scored for the fourth straight game against the Blue Jackets.
The Wild had several opportunities to win it in overtime, but were turned away by Garon.
NOTES: Wild D Marco Scandella broke his finger in the second period and did not return. ... North Dakota hockey coach Dave Hakstol was booed after being introduced before the game. Hakstol's team faces Denver in the WCHA Final Five on Saturday night at the Xcel Energy Center. The hometown Minnesota Gophers did not qualify for the Final Five. ... Columbus has not scored a power-play goal against the Wild in its last 32 chances.