Luongo stopped 17-of-18 shots in the first period and all 11 he saw in the final frame, allowing the Canucks to use a late goal from Kevin Bieksa to force extra time.
Ryan Kelser also scored in regulation for the Northeast Division-leading Canucks, who began a four-game road trip with a victory.
David Jones and Jay McClement scored for the Avalanche, who fell to 7-1 in shootouts for the season.
Jean-Sebastien Giguere turned aside 27-of-29 shots for Colorado.
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