Byfuglien corralled the puck at center ice, moved to the point and lifted a long shot that eluded the Maple Leafs' goaltender, handing the Jets their fifth victory in six games.
Mark Scheifele and Zach Bogosian also scored for Winnipeg.
Ondrej Pavelec earned the win with 24 saves.
Phil Kessel recorded a goal and an assist for Toronto, which dropped a second straight after a six-game winning streak.
Bernier stopped 12-of-14 shots after replacing starter James Reimer, who let in two goals on 14 shots.
2014: The Year in Music [PHOTOS]