Brunette took a pass from Brett Clark and deked goaltender Roberto Luongo, who fell for the fake to the left, allowing Brunette to slip the puck past him on the glove side to break a 2-2 tie at 3:22.
The Avs had ended the second period on a down note after failing to capitalize on a 5-on-3 power play, missing three shots despite having the two-man advantage for a full two minutes. Lukas Krajicek was called for hooking, and Rory Fitzpatrick was caught slashing by the time play stopped.
Colorado's Jose Theodore made 36 saves as the Canucks outshot the Avalanche 38-20.
The Avs entered the game averaging 35 shots per game, tops in the NHL. But they managed just six in the first period, when the Canucks had 15, including Markus Naslund's goal with 53 seconds left.
Alexandre Burrows had just come off the bench when he stole the puck from Paul Stastny at the blue-line and took it down the ice for the pass to Naslund that gave Vancouver a 1-0 lead.
Colorado tied it 92 seconds into the second period when Clark took a pass from Joe Sakic and slapped a shot from the high slot that deflected off Brunette's stick past Luongo. The goalie jumped up and appeared to campaign for interference.
Daniel Sedin's power-play goal at 6:32 of the second period gave Vancouver a 2-1 lead, but the Avalanche tied it again on Stastny's goal at 12:46. Stastny poked the puck past Luongo after a flurry in front of the net.
Notes: Colorado's 51 wins against the Canucks are the second most against an opponent in franchise history behind the 65 victories they've recorded over the Hartford/Carolina franchise. ... Sakic has 88 points in 67 career games against the Canucks, his highest scoring total against an opponent. It's also the most points the Canucks have allowed to one player.