Toronto held Boston to just six shots in the third period, but Boston got a goal when Bergeron scored at 10:36.
Lucic had pulled the Bruins within 2-1 just 1:48 after Kulemin's breakaway goal gave Toronto a 2-0 lead at 8:01 of the second.
Lucic took a pass from Rich Peverley after a turnover in the Leafs' end and went past three Toronto players at the blue line to get in alone on James Reimer and slip the puck between his pads.
"I thought we limited their quality chances, but we gave up a few that cost us," Toronto defenseman Dion Phaneuf said.
Bergeron tied it 2-2 on a backhand off a pass from Dougie Hamilton. The rookie defenseman cycled behind the net and found Bergeron alone, carried the puck behind the net and spotted Bergeron alone low in the right circle.
Joffrey Lupul scored on Toronto's first power play, giving the Leafs a 1-0 lead. Kulemin put the Leafs up 2-0 on a breakaway with 11:59 left in the second. Kadri blocked a shot, controlled the puck and spotted Kulemin streaking up the ice. The assist gave Kadri a team-high 35 points.
Notes: Lupul was returning from a two-game suspension for an illegal check to the head of Tampa Bay defenseman Victor Hedman. ... The Bruins will play their fourth straight Northeast Division opponent Wednesday when they host Montreal. ... Hamilton picked up his 10th career assist on Bergeron's goal in the third.