Fisher backhanded a loose puck 42 seconds into the net to break a 3-3 tie Winnipeg had forged late in the second period on goals scored 27 seconds apart by Mark Scheifele and Blake Wheeler.
Carter Hutton made Fisher's goal stand up by making 11 of his 24 saves for the game in the third period.
Seth Jones, Nick Spaling and Roman Josi also scored for Nashville.
The Jets had a goal and an assist from Scheifele. Chris Thornburn also scored, and Ondrej Pavelec tallied 21 saves.
The loss was only the second in eight games since Paul Maurice took over as Winnipeg's head coach on Jan. 12.