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 and earned his first win since Nov. 27.
"He made some timely saves when we needed him to and he was calm back there," Nashville coach Barry Trotz said. "From our standpoint, it was a combination of solid goaltending and a desperate competitive team going against a fresh, veteran team."
Shea Weber led Nashville in ice time with 29:52 and posted an assist.
"This was a huge game for us," Weber said. "We have been on a tough stretch this past year, but hopefully we can get it right now."
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. ... Nashville rookie D Seth Jones had two assists for his first multipoint game in the NHL.