The Canadiens owned a 3-0 lead early in the second period, but could not produce another goal against Roberto Luongo in regulation, the overtime or the shootout.
Hodgson's goal came in the first round of the shootout. The Canucks won their third straight and for the eighth time in nine games.
Luongo made 20 saves in the victory. Price stopped 28-of-31 shots.
Montreal has lost six out of seven, but each of its last three losses have been in overtime.
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