Burrows sent a perfect pass into the crease that Daniel Sedin tipped in with 44 seconds left in the second period, putting the Canucks ahead for good at 3-2.
The Vancouver winger then gave the Canucks a 4-2 advantage with 6 minutes left in the final frame -- a tally that proved to the game-winner after Toronto's Phil Kessel scored his 19th goal 3 minutes later.
Roberto Luongo turned aside 26-of-29 shots for Vancouver, who beat the Maple Leafs for the ninth straight meeting.
Jonas Gustavsson made 31 saves for Toronto.
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