Greaves stopped Stars forward Mavrik Bourque on a breakaway at 5:16 of the third period to preserve the lead, and Dallas forward Justin Hryckowian hit the crossbar at 18:51 from in tight.
“It's frustrating,” Stars forward Wyatt Johnston said. “You’ve got to work hard to get your looks and then you also got to work even harder to finish them and continuing to get looks.”
Dallas has scored one goal in each of its past four losses, with a 6-2 victory against the Boston Bruins serving as the only win during the stretch.
“A lot of good things, just didn't get the win,” Stars defenseman Thomas Harley said. “I thought we created some offense tonight. I thought we defended reasonably well but we just couldn't get one over the line.”
The Blue Jackets are 4-1-0 since coach Rick Bowness replaced Dean Evason, who was fired on Jan. 12.
“The fans got their money's worth tonight, for sure,” Bowness said. “Actually, that's how we want to play. And that's fun, for me to be behind the bench watching that effort.”
Columbus forward Charlie Coyle played his 1,000th NHL game.
“It's just nice to win. It's a big two points, but these milestones and things to celebrate are all fun,” he said. “It's a part of it. It's nice but when you win, it makes it that much better and you can enjoy it a little more. So that's what I'm happy about.
“I know we're all happy about that. That was a great game by us to get the job done. That was huge.”
NOTES: Werenski recorded his 18th career game-winning goal and passed David Vyborny (17) for the fifth-most in franchise history. ... Blue Jackets forward Mason Marchment returned to the lineup after missing the past eight games with an upper-body injury. He had one shot on goal in 18:23 of ice time. ... Columbus forward Isac Lundestrom returned after missing 13 games with a lower-body injury. He had one assist in 7:27 of ice time.