The Predators won at home for the first time in four tries this season.
Nashville then regained control by scoring three times in the second, getting goals from Patric Hornqvist and Nick Spaling before Fisher put a slap shot past Mathieu Garon.
Stamkos scored his second of the game in the third period, but Pekka Rinne held on to get the win while making 29 saves.
The two goals for Fisher were his first of the season.
Tampa Bay had a three-game winning streak snapped. Garon gave up five goals on 17 shots before being removed in favor of Dwayne Roloson, who stopped all 10 shots he faced.