Kyle Wellwood, Ryan Johnson, Pavol Demitra and Daniel Sedin also scored for the Canucks, who are 7-1-2 in their last 10 games. Mattias Ohlund collected three assists, while Taylor Pyatt and Jannik Hansen both recorded two helpers.
Roberto Luongo made 39 saves to notch his 11th win of the season.
Chris Drury registered a goal and an assist for the Rangers, who had a three- game winning streak snapped. Henrik Lundqvist allowed five goals on 17 shots before he was yanked in the second period. The Swedish netminder had been nearly unbeatable recently, yielding two goals or less in his last 12 appearances entering Wednesday.
Stephen Valiquette stopped all 15 shots he faced.
Forward Markus Naslund, who joined New York in the offseason, picked up an assist in his first game against his former team. Naslund spent 12 years in Vancouver and was the team's captain from the start of the 2000-01 campaign until his departure over the summer.
Vancouver grabbed a 1-0 lead 4:16 into the contest on Wellwood's team-best eighth goal of the season.
The Canucks struck again while shorthanded. New York defenseman Wade Redden coughed the puck up at the left point and Hansen fed Burrows, who skated in alone and snapped a wrister that beat Lundqvist on the blocker side midway through the first period.
Burrows put Vancouver up 3-0 early in the middle frame, rifling a shot from down low that found the back of the net at the 1:23 mark. The Canucks took a four-goal lead 58 seconds later, as Pyatt shoveled the puck from behind the net to Johnson and he was able to sneak it past the New York netminder.
Drury's power-play marker put New York on the board 51 seconds after Johnson recorded his first goal in a Vancouver uniform.
A power-play tally by Demitra made it a 5-1 game at 5:33 and chased Lundqvist from the crease.
Aaron Voros lit the lamp three minutes later for the Rangers, and Ryan Callahan scored on the power play at the 7:45 mark of the third, trimming the deficit to 5-3.
Sedin added an empty-net marker in the final seconds.