St. Louis and Dallas were a combined 1-for-12 on the power play.
Jaden Schwartz scored in regulation for the Blues (33-17-9), and Brian Elliott made 17 saves to improve to 10-3-3 since Jake Allen went down with a knee injury on Jan. 8.
The Blues are 4-0-1 in their past five games.
"I think we're trusting our work and it shows up on [penalty kill], it shows up in 5-on-5," St. Louis coach Ken Hitchcock said. "When you trust your work, whether it's 1-0 or 3-0, you feel comfortable. I think that's what's happened right now. We have people demanding a lot of themselves. We have more to give, especially up front; we have some people who have to get up to speed up front, but what we're doing as a group is trusting our work, and that's allowing us to stay really competitive in the games right now."
Mattias Janmark scored for the Stars (37-15-6), who were held to a season-low 18 shots and had their four-game winning streak end. It was Dallas' first loss in six road games.
"I think when you see a game like that, you see that it was two teams that played sound defensively," Stars coach Lindy Ruff said. "They did their job in the defensive zone. Overall, it was two good teams going at it."