Nash broke out of a slump with two goals and two assists, 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.
"It just seems like it was going in today," Nash said. "I've been getting a ton of chances. I've been hitting some posts and hitting some chests of the goalies, but tonight it went in."
The same could be said about the Blue Jackets, who won for just the second time in their last 11 games.
"All those pucks have been hitting sticks and crossbars and goalie's back ends," Columbus coach Scott Arniel said. "Finally a misplay by them and we were fortunate enough to have guys there ready to bury it."
Burns made no excuses after the game. He said he was trying to pass the puck to teammate Mikko Koivu.
"I fanned on it or missed it. It's a bad bounce," he said.
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. The Wild entered Saturday in 11th place in the Western Conference while Columbus was in 12th.
But scoring chances were not hard to come by on Saturday.
"You know that it was two teams that were desperate for two points and were going to throw everything at each other," Arniel said. "That's probably what opened things up a little bit."
Columbus managed only 22 goals in its previous 10 games, but appeared to get a lift near the end of the second period.
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.
"It was a good feeling. And I think pretty much the whole overtime we were pushing and finding scoring opportunities," Mittinen said. "Unfortunately a mistake ends up in our net. Not a good way to finish the game."
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.
"He made some outstanding saves there in overtime," Arniel said. "He definitely gave us good enough goaltending to win."
- Wild D Marco Scandella broke his finger in the second period and did not return.
- Columbus D Grant Clitsome left in the second period after a knee-on-knee collision with Koivu. Clitsome will be evaluated by Columbus doctors on Sunday.
- 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.
- Madden had two assists for the first time in over a year.