Fisher's first goal came on a power play with 3:18 left in the third period and gave the Predators a 2-1 lead.
But with 36.7 seconds to go, the Kings notched an equalizer when Jeff Carter unleashed a quick shot from the right circle off a pass from Anze Kopitar that eluded Nashville goalie Marek Mazanec.
The game seemed destined for overtime but Nashville answered just 10 seconds later. Fisher's shot from the right side hit Kings goalie Ben Scrivens and slowly inched over goal line before the goaltender could get a glove on it.
A video review upheld the ruling on the ice and the Predators held on to break a five-game losing streak.
Fisher also had an assist and Patric Hornqvist added a goal and an assist for Nashville.
Mazanec stopped 26-of-28 shots.
Kopitar provided a goal and an assist and Scrivens made 28 saves for the Kings.