It was Vancouver's 10th straight win over Toronto, covering a span of more than eight years.
Manny Malhotra, Sami Salo and Kevin Bieksa also scored for the Canucks (37-15-6), who posted their sixth win in seven games.
Dion Phaneuf and Nikolai Kulemin, with 19.2 seconds left in the game, replied for the Maple Leafs (29-24-6), who suffered their sixth loss in seven outings.
The Canucks led 2-0 after a first period in which they outshot the Maple Leafs 11-4.
Burrows opened the scoring 7:49 into the game on a 2-on-1 with Daniel Sedin against Phaneuf. Burrows gave a lead pass to Sedin who delivered a nice flip pass over the Toronto captain's stick and past Toronto goalie James Reimer.
Malhotra gave the Canucks a 2-0 lead a minute and 27 seconds later as he dove to tip in Max Lapierre's pass from the side boards. It was only Malhotra's second goal and third point in 14 games.
The Caucks came close to increasing their lead with just under two minutes remaining in the first, but Reimer robbed Henrik Sedin as he tried to put in a rebound from the edge of the crease.
Ron Wilson on the Leafs' defensive troubles:
"You can't fault James (Reimer) ... They had a number of two-on-ones when we pinched. Those are things you talk about before the game, knowing who you're playing against: No turnovers. They don't need a lot to make you pay and tonight they made us pay."
Burrows put the Canucks ahead 3-0 at 3:18 of the second period when he took a pass from Henrik Sedin and wristed home a shot while left alone in the slot.
But Phaneuf put Toronto on the scoreboard just under three minutes later as he one-timed a Phil Kessel cross-ice pass behind Roberto Luongo.
Moments later, the Canucks goaltender made a diving blocker save on Joffrey Lupul to maintain Vancouver's two-goal advantage.
Then Luongo foiled Kessl on a rebound during a Toronto power play, gloving the puck after the Leaf fanned on the shot. Kessel slammed his stick against the boards in frustration.
Luongo finished the game with 24 saves on 26 shots.
Daniel Sedin put the Canucks ahead 4-1 midway through the second period when he scored on a pass from the corner boards by Henrik.
That chased Reimer and brought in Jonas Gustavsson in the Toronto net. Reimer allowed four goals in 18 shots.
Gustavsson let in the first shot he faced. Salo one-timed a Daniel Sedin shot past the screened netminder while on the power play.
Henrik Sedin picked up his fourth assist of the night at 9:09 of the third period as Jannik Hansen re-directed his pass to Bieksa
Vancouver converted one of two power-play opportunities. Toronto was blanked on two.
Notes: Canucks centre Cody Hodgson, a Toronto native, celebrated his 22nd birthday Saturday. ... Canucks D Keith Ballard missed his fifth straight game with a neck injury that is also being investigated for a concussion. ... Leafs D Luke Schenn returned to the lineup after being a healthy scratch during Wednesday's overtime win in Edmonton. He assisted on Kulemin's late goal. As a result of Schenn's return, rearguard Mike Komisarek went back to the press box after playing in Edmonton and Calgary. ... Injured Toronto D Carl Gunnarsson (ankle) has resumed skating on his own. ... Late in the game, a fan threw a dead salmon on the ice.
Tuesday, February 21
vs. New Jersey Devils
7PM ET (Sportsnet, AM640)
The Leafs return home to begin a four-game homestand against the Devils who are 3-1-1 in their last five games.
New Jersey is on the road Sunday as they face Montreal. The Devils currently sit six points ahead of the Leafs in the Eastern Conference.