Kane gave the Blackhawks a lead they wouldn't relinquish when he scored with 2:33 remaining in the first period, putting in shot by Johnny Oduya that had deflected off a defenseman's stick.
He doubled the advantage 5 minutes into the middle frame when he converted a pass from Bryan Bickell for his ninth tally of the playoffs.
Dave Bolland added an empty-netter for Chicago, which notched a second straight victory following Wednesday's dramatic 6-5 overtime win in Boston.
Corey Crawford turned aside 23-of-24 shots, including 8-of-9 in the third period.
Zdeno Chara scored early in the final frame, cutting the deficit to one goal, but the Bruins could get no closer.
Tuukka Rask made 24 stops for Boston, which will try to even the series at home in Monday's Game 6.
Bruins center Patrice Bergeron was hospitalized after leaving the game in the second period.
The team announced Bergeron, who has nine goals and six assists in 20 playoff games, was "taken to a local hospital in Chicago for observation."