Right after the opening faceoff, Burrows received a pass from Daniel Sedin and streaked down the boards, eluding Boston captain Zdeno Chara behind the net before extending his stick for a wraparound goal to beat Boston goalie Tim Thomas for his second score of the game.
Daniel Sedin tied it with 10:23 left in the third period for the Canucks, who are halfway to the franchise's first championship. Roberto Luongo made 28 saves for the Canucks.
Burrows, who avoided suspension for this game after being accused of biting Boston's Patrice Bergeron in Game 1, also up Sedin's tying goal with a sharp pass from the slot.
Game 3 is Monday in Boston.
Thomas stopped 30 shots for the Bruins, but his aggressiveness put him out of position during the brief overtime. Burrows used his speed to elude the 6-foot-9 Chara before slipping the puck into Thomas' unguarded net.
Mark Recchi put the Bruins ahead in the second period with a power-play goal just 2:35 after Milan Lucic scored Boston's first goal of the finals. Recchi, the NHL's oldest active player at 43, ended the Bruins' 5-for-68 power-play drought with a deflection from the slot.
The Canucks got a pregame boost from the inspirational return of center Manny Malhotra, who hadn't played since incurring a career-threatening eye injury on March 16. But until Daniel Sedin's late goal, Boston played a second straight game of impressive shutdown defense against Vancouver's vaunted top line featuring the NHL's last two scoring champions.
Turns out the Bruins should have been paying attention to Burrows, their less-heralded linemate who has 17 points in the postseason.