Mason, who came into the game with a 3.49 goals-against average, won for only the ninth time in 32 decisions. He made 32 saves and slowed down Colorado's bid to make the playoffs.
The Avalanche missed a chance to move past Dallas into the eighth and final post-season position in the Western Conference.
It was just the 19th win of the season for Columbus. Every other team in the league has at least 24 victories.
Nikita Nikitin scored on the power play in the first period and Fedor Tyutin put a slap shot past Semyon Varlamov in the second. Varlamov stopped 26 shots and suffered his first loss since Feb. 19.