Vancouver prevailed in a rematch of last year's Stanley Cup finalists that featured 110 minutes in penalties.
Boston's Tyler Seguin drew a tripping penalty with 4 minutes left in the second period and Burrows made the Bruins pay, notching the equalizer at 2-2.
Then with 1:30 left in the middle frame, Shawn Thorton was whistled for a 5-minute major clipping penalty and the Canucks scored twice, using tallies by Henrik Sedin and Cody Hodgson to grab control of the game.
Hodgson also logged an assist, Ryan Kesler scored and Cory Schneider turned aside 36-of-39 shots for Vancouver, which lost in seven games to Boston in the 2011 Stanley Cup Finals.
Brad Marchand scored and drew a game misconduct for Boston, while Rich Peverley and David Kreji also found the net in the loss.
Tim Thomas made 32 saves for the Bruins.