The Bruins go on to face the winner of the Montreal-Washington series, to be decided in a seventh game Wednesday.
Milan Lucic added a pair of assists and goalie Tuukka Rask earned his first career post-season series after turning aside 27 shots.
Sabres' leading scorer Thomas Vanek returned from an ankle injury after missing three games to notch a goal with 1:13 left in the regulation. The tally pulled Buffalo to within 4-3, but that was as close as the Sabres would come.
Patrick Kaleta and Nathan Gerbe got Buffalo's other goals while Ryan Miller allowed four goals on 32 shots.
Swim Week Miami Beach 2014 [PHOTOS]