Habs will play seven
The Bruins, who lost all 20 previous best-of-seven series in which they trailed 3-1, will try to complete their bid to eliminate the top-seeded Canadiens in Montreal on Monday night.
Sturm scored after rookie goalie Carey Price stopped his first shot but failed to control the rebound. Sturm collected it, skated across the crease from left to right, and a sprawling Price couldn't get to the other side in time.
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So the Bruins still have a chance to win a playoff series for the first time in nine years and complete an improbable comeback after they began the series seeded eighth in the Eastern Conference.
They took their first lead of the game 4-3 with 4:15 left in the third period on Phil Kessel's second goal of the game.
But it lasted just 11 seconds until Christopher Higgins scored his second of the game
Five of the six games in the series have been decided by one goal. And Price, outstanding in the first four games, allowed four goals in the third period for the second straight game. On Thursday night, Boston won 5-1 after the score was tied 1-1 after the second period.
Saku Koivu, who missed the last nine games with a broken bone in his left foot, set up the first goal when he won a faceoff. Higgins then got the puck away from defenseman Dennis Wideman and scored at 9:44 of the first period.
Kessel tied it with some nifty stickhandling. About 30 feet from the goal, he nudged the puck between the legs of Francis Bouillon, picked it up on the other side, and tied the game at 1:54 of the second.
Montreal took a 2-1 lead on a breakaway goal by Tomas Plekanec at 7:43 of the second just eight seconds after he left the penalty box.
But Boston tied it 3:13 into the third period. Vladimir Sobotka took Peter Schafer's pass to the right the net, skated across the crease and put a backhander by goalie Carey Price.
The Canadiens went ahead again on Bouillon's shot from the left point that deflected off the stick of Boston defenseman Shane Hnidy and past goalie Tim Thomas at 10:04 of the third period.
And the Bruins came back again to tie it at 3-3 just two minutes later when Milan Lucic redirected Aaron Ward's shot from the right point and into the net.
Notes: Ward returned after missing one game with a knee injury. ... In 2004, current Bruins coach Claude Julien led Montreal back from a 3-1 series deficit in the first round to beat Boston.