In a contentious battle that saw two game misconduct penalties handed out, the Avalanche had to sweat out the victory after taking a 4-1 lead into the third period.
Zach Parise and Mikael Granlund then scored for Minnesota to bring the Wild within one with 7:09 still to play.
The Avalanche survived, however, to win for seventh time in nine games. Colorado moved within six points of first-place Chicago in the Central Division.
Varlamov made 31 saves to get the shaky win. Darcy Kuemper was removed from the Minnesota net after giving up four goals on 23 shots. The winning goal came against Niklas Backstrom.
Minnesota's Matt Cooke and Colorado's Nate Guenin received the game misconduct penalties as well as roughing penalties for their confrontation early in the third period.