New York's Henrik Lundqvist and the Sabres' Ryan Miller posted a double shutout through 65 minutes before the Rangers added to the largest point total in the Eastern Conference.
Callahan put a wrist shot past Miller after Lundqvist stopped Ville Leino to open the fifth round.
Lundqvist, third in the NHL this season in both goals-against average and save percentage, recorded his sixth shutout of the season and the 41st of his career. He made 34 saves.
Miller stopped 29 shots in what was Buffalo's first loss in three games.
Celebrity Families of 2014 [PHOTOS]