Werenski and Max Domi each had a goal and an assist, and Jack Roslovic had three assists for the Blue Jackets (11-12-6), who had lost three straight. Elvis Merzlikins made 27 saves.
Werenski, who led NHL defensemen with 20 goals last season, scored his second of the season on the rebound of Oliver Bjorkstrand's shot.
"He shot it, and I just tried to follow the rebound. Not much to it," Werenski said. "It wasn't the best game, but we found a way to win. That's the important thing."
Andrew Cogliano had a goal and an assist, and Jake Oettinger made 22 saves for the Stars (8-9-6), including on a penalty shot by Emil Bemstrom with the game tied 3-3 at 9:02 of the third period.
Merzlikins saved Jason Robertson's backhand from in close at 2:53 of overtime.
"When you get those chances in overtime you hope maybe it would go in," Dallas coach Rick Bowness said. "We had some unreal looks at the end of that third period. It's unreal the chances we had and didn't score."
The teams will finish the back-to-back set here Sunday (5 p.m. ET; FS-O, FS-SW, NHL.TV).
Domi gave Columbus a 1-0 lead at 9:07 of the first period. The forward has scored five goals this season and four points (two goals, two assists) in the past two games.
"I think the last five, six I have felt like myself," Domi said. "We found a way to win; that's what it is all about. We knew we had to be better. We took it upon ourselves. No one is going to come and save us.
"Anytime you win, no matter what team you are, where you are in [the] standings, what time of year, a win is a win. That is why we play hockey. No one wants to lose. This was a huge two points tonight. Get ready to go for tomorrow and hopefully do the same thing."