Varlamov has replaced Jean-Sebastien Giguere as Colorado's front-line goaltender and has given up no more than one goal in four of his last five starts.
He made 28 saves Friday in his seventh shutout in four NHL seasons.
David Jones tallied twice to pace the Colorado attack, opening the scoring with 26 seconds left in the first period and producing the Avalanche's third goal 1:35 into the final stanza.
Colorado began the night two points out of the final playoff spot in the Western Conference and is one of five teams battling for that position. That group opened the evening separated by just three points.
The Blue Jackets went into the game with the third-fewest goals in the league (142), but were shut out for only the third time this season. They have the worst record in the NHL with 18 victories in 61 games.