Leino used a heavy wrister to beat Cory Schneider before Nathan Lieuwen made a right-pad save on Mike Sislo's backhand attempt for the win.
Tyler Ennis scored twice while Lieuwen grabbed his first NHL win thanks to stopping 33 shots for the Sabres, who snapped a four-game slide.
New Jersey's shootout woes continue, as the team has dropped 15 straight games decided in such a fashion, with 11 coming this season.
The Devils gave up a big point, as the team is three points back of Columbus for the second wild card spot in the Eastern Conference with just six games to play.
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