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.