The Avalanche faced a 3-1 deficit going into the third period, but Chris Stewart scored his 11th goal of the season and Paul Stastny put a slapshot past Antero Niittymaki with 5:11 to go in regulation to tie it.
Colorado's other goal came from rookie defenseman Kevin Shattenkirk, the first of his NHL career. Peter Budaj made 40 saves to get the win in his 10th straight start in place of the injured Craig Anderson.
Anaheim, which had won three in a row, built its lead on goals from Logan Couture, Dan Boyle and Mike Moore. Niittymaki made 18 saves.