In the extra session, Matt Duchene circled around the left boards to get clear of defenders before firing a pass to the front of the net where Barrie tipped it home for the win.
Duchene had a goal and an assist while Gabriel Landeskog also had a goal for the Avalanche, who have won their past two after a three-game losing streak. Semyon Varlamov stopped 29 shots in the win.
With 98 points, Colorado sits just one point back of Chicago for second place in the Central Division with a game in hand.
Yannick Weber and Zack Kassian scored for the Canucks, who had a three-game winning streak stopped. Eddie Lack turned away 25 shots in the loss.
Vancouver's playoff hopes are quickly fading, as the team is five points back of Phoenix for the final wild card spot with just seven games to play.