Varlamov has been forced into action due to the groin injury suffered by regular goaltender Jean-Sebastien Giguere, and his excellent outing against the Oilers snapped Colorado's two-game slide.
The only goal against Varlamov came in the second period by Ryan Jones after the Avalanche had built a 3-0 lead.
Erik Johnson scored in the first period and Colorado got second-period goals from Mark Olver and Gabriel Landeskog.
Olver made his season debut after being called up from Lake Erie of the American Hockey League. He had two previous NHL goals while appearing in 18 games for Colorado last season.
Devan Dubnyk stopped 20 shots for the Oilers, who have lost four out of five.