The teams combined for three goals in the final 3:18
Fisher put Nashville ahead 2-1 with 3:18 left with a slap shot from the high slot that beat goalie Ben Scrivens low to the stick side.
Carter tied it with only 36.7 seconds to go, but Fisher put the Predators back in front 10 seconds later when his shot from the left side just made it over the goal line.
Hornqvist had given the Predators a 1-0 lead with 5:18 left in the first.
Scrivens stopped Fisher's shot from the high slot, but Hornqvist — who was standing just outside the crease — backhanded in the rebound for his eighth goal.
Hornqvist hadn't scored a goal in six games.
Kopitar then tied it at 7:05 of the second.
With the Kings on a power play, Kopitar skated to the top of the right circle and beat goalie Marek Mazanec with a wrist shot that was tucked just inside the far post.
Los Angeles hadn't scored on the power play in its previous five games. Mazanec finished with 26 saves.
NOTES: Kopitar has a three-game point streak. ... The Kings have allowed at least one power-play goal in three consecutive games. ... It was Fisher's first two-goal game of the season.