The Ducks, who came into the game with the second-best mark in the Western Conference, have now lost three straight games. They also haven't scored in two contests - going 139 minutes 16 seconds without a goal.
Paul Stastny added a goal for the Avalanche, who won for the second time in six games.
Jean-Sebastien Giguere finished with 27 saves.
Andrew Brunette, Joe Sakic and Jordan Leopold all had assists for Colorado.
Budaj was back in net after being pulled in the second period against Dallas on Sunday. Budaj gave up four goals before Jose Theodore relieved him.
But Budaj came up with one big save after another against a Ducks team that was second in the West in scoring at 3.21 goals a game. It was Budaj's fourth career shutout.
He beat Columbus 3-0 on Nov. 17, stopping 27 shots.