Michael Hutchinson, who was drafted by the Bruins in 2008, made his second NHL start and stopped Brad Marchand with his blocker on Boston's final try to get his first win. He had 32 saves in regulation and overtime.
Kane tied the game with 1:57 remaining in the third period when he scored on a wrist shot from the left circle off a drop pass from Blake Wheeler, and Boston failed to capitalize on a 40-second 4-on-3 in overtime after Olli Jokinen went off for holding.
Marchand scored Boston's goal midway through the first period when he collected a rebound to the right of Hutchinson after Reilly Smith flung the puck off the end boards from mid-ice and beat the netminder with a deke.
Johnson had 36 saves for the Bruins, who moved three points ahead of idle Anaheim for the NHL points lead.
The Bruins, who have already locked up the Eastern Conference's No. 1 seed, are trying to win the Presidents' Trophy for the first time since the 1988-89 season.
But they have just one win in their last five games (1-1-3) on the heels of a 15-0-1 stretch.