Halak registered his second shutout in three games, and third of the season, while improving to 9-0-3 in his last 12 decisions, as the Blues (27-12-6) grabbed a share of the overall NHL lead with 60 points.
Oshie broke a scoreless tie 11:26 into the final period after crashing the net and putting the puck past Stars goalie Kari Lehtonen.
St. Louis won for the 19th time in 25 home games.
Lehtonen finished with 31 saves Dallas, which was missing leading scorer Jamie Benn.
The Stars forward is out for at least 10 days after undergoing an emergency appendectomy Sunday.
2014: The Year in Music [PHOTOS]