The teams made up for the lack of first-period scoring during the second with two goals each, including a pair in the last 27 seconds that made it 2-2.
With the Jets crowding the Boston net, the rebound from Zach Bogosian's point shot was flipped into the net by Burmistrov at 1:43 of the second period.
The Bruins evened the score at 1 with some help from Hainsey.
Seguin was credited with a deflected goal after Boston captain Zdeno Chara's shot from the point eventually bounced into the net off Hainsey's leg at 10:57.
Kane gave the Jets a 2-1 lead with 27 seconds left in the period when Nik Antropov sent a backhand shot from the backboards to the front of the net and Kane snapped in his own rebound.
But with just two seconds left on the clock, Paille deflected a long shot from Johnny Boychuk past Pavelec that tied the game at two heading into the final period.
Winnipeg pressed for the tying goal in the dying seconds, but Rask kept the puck out as Boston won the second game of a five-game road trip.
Notes: The Bruins were without F Milan Lucic, who arrived in Winnipeg on Saturday but left the same day to return to Boston for what the team called personal reasons. ... Veteran F Jay Pandolfo, 38, was playing his first game with the Bruins after signing a one-year contract as a free agent on Feb. 12. ... Jets D Mark Stuart was playing in his 400th NHL game. His first 283 were with Boston.