Werenski tied it 1-1 with his first goal of the season on a backhander over Casey DeSmith's left pad after skating in from the left face-off circle.
Valeri Nichushkin and Kent Johnston scored in the shootout for Columbus.
Cam Talbot made 27 saves for the Blue Jackets (7-7-0), who have won two straight.
Mikko Rantanen scored, and Casey DeSmith made 29 saves for the Stars (9-5-2), who are 2-2-1 in their past five games.
Rantanen scored for the third straight game to put Dallas up 1-0 at 6:27 of the third period.
After Ivan Provorov passed the puck into the slot, Wyatt Johnston pounced on it and was stopped by Talbot. The rebound went to the right of the net where Rantanen collected it and calmly waited for an opening at the near post.
