"It's what we need to get over a slump like this," Wilson said. "You need the big players to make big plays, and that's what he did. He was our biggest player and that was one of the biggest plays of the year for us."
Dallas' six-game winning streak was snapped.
Nash, who also scored in the shootout for Columbus, tied the game with 46.1 seconds left with his 14th goal.
"It was big," Nash, the Blue Jackets captain, said of the victory. "We'd been in a bit of a losing streak here. It's been tough to find goals."
Steve Ott and Brad Richards had goals for the Stars, who were trying to match the franchise record of seven consecutive wins.
After Nash and Richards scored in the shootout, Wilson went third and used a hard forehand to beat Lehtonen on the stick side.
Mathieu Garon then made a right toe save on Jamie Benn to seal the win.
Kris Russell also scored for the Blue Jackets. Garon made 16 saves through overtime and two more in the shootout.
"I knew he was a good shooter, so I really wasn't surprised to see him there," Garon said of Wilson. "He took the same shot as Nasher took. It was good to see that."
Ott had jammed in a perfect feed from Trevor Daley with 7:52 left in regulation to break a 1-1 tie.
It stayed that way with Lehtonen, who had 27 saves. He made several fine stops until Nash rifled a low wrister through traffic provided by R.J. Umberger in the final minute. The puck slipped over Lehtonen's right leg pad.
The Blue Jackets had the best of the overtime, taking all four shots, but Lehtonen was up to the task.
Wilson, one of the early-season surprises for Columbus, was moved up to center the second line in an attempt to shake up the Blue Jackets' moribund offense. It had been outscored 12-2 in the last two losses.
Despite their recent troubles, the Blue Jackets are off to their best start (15-10-1) in the franchise's 10 seasons.
"That was a big two points that we needed," Blue Jackets coach Scott Arniel said.
The Blue Jackets, 10-1 when leading after a period, went ahead early when Russell took a pass from Jake Voracek and surprised Lehtonen by backhanding a shot high over his stick side.
The Stars tied it at 15:45 of the second after dominating for most of the period, as Richards scored his 12th goal with a wrister from the slot. Garon got a glove on the shot, but it slipped in behind him.
"We lost in a shootout," Dallas coach Marc Crawford said. "There's going to be somebody who's disappointed and somebody who's elated after a shootout. We're the disappointed ones. I liked our team. I thought we were really good in the second. I thought we were the better team in the third period. We got a bad break against us."
Material from wire services and team media was used in this report.
1 - 0 CBJ
1 - 1 Tie
2 - 1 DAL