Detroit hadn’t lost at Joe Louis Arena since Nov. 3 against Calgary.
The NHL-leading Red Wings hold a one-point edge over Vancouver in the Western Conference.
A sold-out crowd stood during the shootout, which started with Canucks goalie Roberto Luongo stopping Jiri Hudler’s shot. Detroit’s Jimmy Howard then went low to smother David Booth’s attempt.
Henrik Zetterberg missed the net on Detroit’s second attempt, and Alex Edler was then denied by Howard.
Todd Bertuzzi, who signed a two-year extension with the Red Wings earlier in the day, couldn’t put Detroit ahead. Burrows then won it with a backhanded shot.
DUCKS 3, HURRICANES 2 (SO)
In Raleigh, N.C., Saku Koivu scored the only goal in the shootout to lift Anaheim over Carolina.
Corey Perry, who reached 30 goals for the third time in his career, and Luca Sbisa scored in regulation for Anaheim. Jerome Samson and Jamie McBain had goals for Carolina.
Jonas Hiller had 30 saves for the Ducks.
WILD 3, PANTHERS 2 (SO)
In Sunrise, Fla., Matt Cullen and Erik Christensen scored in the shootout in Minnesota’s victory against Florida.
Cullen and Kyle Brodziak scored in regulation, Dany Heatley had two assists, and Niklas Backstrom stopped 41 shots for the Wild.
SHARKS 2, MAPLE LEAFS 1
In Toronto, Patrick Marleau scored two goals, and San Jose sent stumbling Toronto to its seventh loss in eight games.
Jake Gardiner scored for the Maple Leafs, who are 1-6-1 since Feb. 7.
BLUES 3, PREDATORS 2 (SO)
In Nashville, shootout goals by T.J. Oshie and Andy McDonald pushed St. Louis past Nashville.
Chris Stewart and Vladimir Sobotka scored for St. Louis in regulation.
Roman Josi and Colin Wilson had goals for Nashville.