Patrik Berglund added a pivotal insurance goal early in the third period as the Blues won their fourth straight.
Colorado finally solved Miller midway through the final period when P.A. Parenteau lifted a shot over him while he was trying for his first shutout in nearly two years. Miller is 4-0 since being acquired from Buffalo.
Avalanche forward Nathan MacKinnon had his point streak snapped at 13 games. It is the NHL's longest streak by an 18-year-old player.
Miller withstood a furious rally by the Avalanche in the final two minutes after they pulled goalie Semyon Varlamov.
The division-leading Blues are five points ahead of Colorado.
St. Louis also improved to a league-best 17-0-1 in division games. The Blues have outscored the Avs 13-5 in taking all three games in the season series. The teams will meet again on April 5 in St. Louis.
This was a frustrating afternoon for the Avalanche as their four-game winning streak was halted. Early on, they couldn't sneak anything past Miller, even hitting the post twice. They also lost one of their top offensive players when Paul Stastny went out with a back injury after his opening shift.
Berglund gave St.
Moments later, Parenteau beat Miller after receiving a pass from Matt Duchene.
With 6:05 remaining, Gabriel Landeskog had a chance to tie it after getting loose in front, but Miller rose up to stop it.
St. Louis made it 1-0 at 7:03 of the second period on a play set up by a long pass from defenseman Kevin Shattenkirk, who has thrived against his former team since he was traded three years ago.
Deep in his own end, Shattenkirk sent a pass to T.J. Oshie, who, as he was falling down, slid the puck to Backes. From there, Backes guided the puck past Varlamov's left skate.
Shattenkirk has four goals and five assists in 12 games against Colorado.
Both teams had a failed power play in the first period and eight shots. Miller had the save of the period when he stopped a point-blank shot from Parenteau with his left shoulder.
At one point, Colorado's Patrick Bordeleau made his way back to the bench following a big hit, and was pushed by Barret Jackman, then by Ryan Reaves and finally by Maxim Lapierre in a comical scene.
NOTES: Blues coach Ken Hitchcock tied Ron Wilson for seventh place on the NHL career wins list with 648. ... Miller has stopped 84 of 90 shots since his arrival in St. Louis. ... Colorado had 13 of its 27 shots in the final period.
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