In front of a boisterous sell-out crowd of 19,289 at the Pengrowth Saddledome, the two Northwest Division rivals met for the fifth time in 25 days and the result was an emotional and physical game with numerous skirmishes.
With the Canucks leading 2-1, Burrows' beautiful individual effort at 15:29 of the second period gave Vancouver the all-important third goal.
Darting down the wing on a two-on-one, Burrows eluded Mark Giordano's diving swipe at the puck, patiently waited for goaltender Miikka Kiprusoff to commit and go down and then reached back and tucked the puck inside the goal post as he skated past the net.
Roberto Luongo made 27 saves for Vancouver, the biggest being a spectacular glove stop on Byron Ritchie after he was set up for a one-timer by Kristian Huselius on a two-on-one rush with six minutes left in the game.
Trevor Linden and Brendan Morrison also scored for Vancouver (21-18-1). The Canucks won their fourth game in a row, their longest winning streak of the season - with all four wins coming against Northwest Division opponents. The Canucks are tied for first place atop the Northwest with the Minnesota Wild.
Matthew Lombardi scored a pair for Calgary (19-15-4).
The Flames lost for just the second time in the past 14 games at home, both of the losses coming against the Canucks.
Calgary trails the Wild and Canucks by one point.
Vancouver led 2-0 after 20 minutes but Lombardi's two goals helped Calgary cut the deficit to 3-2 after two periods.
Lombardi made it 2-1 six minutes into the second on his fourth short-handed goal of the year.
Sent in alone by Kristian Huselius, Lombardi fended off a diving backcheck by Kevin Bieksa, cut across the front of the Vancouver net and neatly pulled the puck around Luongo. Lombardi is one short-handed goal behind league-leading Pittsburgh rookie Jordan Staal.
After Burrows put Vancouver back in front by two, Lombardi pulled Calgary back to within a goal again at 17:23 when he burst down the wing and whipped a high shot past Luongo from 20 feet out.
Linden opened the scoring at 7:54 of the first period one-timing Morrison's centring pass after Lombardi coughed up the puck behind the Flames net.
Morrison made it 2-0 four minutes later when Sami Salo's shot from the blue-line hit Rhett Warrener's skate and sharply caromed out to Morrison at the side of the scrum, who was left with an open net.
Vancouver finished 1-for-10 with the man advantage while Calgary was 0-for-6.
Kiprusoff had 28 saves for Calgary. The Flames all-star goaltender has given up three or more goals in eight of his past nine starts.
Notes - Calgary D Dion Phaneuf leads the Flames with eight power-play goals. . . The Flames and Canucks meet just twice over the final three months of the season - Feb. 3 in Calgary and Mar. 31 at Vancouver. . . . Vancouver RW Markus Naslund has gone 15 games without a goal. With 100 career power-play goals, he is three away from moving past Tony Tanti into top spot on the Canucks all-time list. . . . Calgary D Andrei Zyuzin was a healthy scratch for the second straight game. . . . Vancouver improves to 16-5-1 when scoring first.