Rask posted his third shutout of the postseason in Game 3 and began Game 4 with a shutout streak of 122 minutes, 26 seconds.
That ended at 129:14 when Handzus scored a short-handed goal at 6:48 of the first period. Then Peverley, who had been struggling offensively, scored his second playoff goal during a rare power-play shift for him at 14:43.
Toews and Kane led the Blackhawks with 23 goals each this season, and neither had scored a goal in the Cup finals. Toews had gone 10 straight games without one, but now has two goals in 21 playoff games.
His fortunes changed when he gave Chicago a 2-1 lead at 6:33 of the second. Michal Rozsival fired a low rising shot from the right point that Toews tipped in near the right post.
Just over two minutes later, Kane made it 3-1. Bryan Bickell shot the puck from low in the left circle. Rask deflected it to the other side and then scrambled to get back in position, but Kane converted the rebound at 8:41.
The Bruins stayed aggressive, and it paid off when Lucic got his sixth goal of the postseason at 14:43. Zdeno Chara passed the puck from the left boards to Lucic in the slot for a backhand from 10 feet past Crawford.