Viktor Arvidsson scored 1:17 into the third to get Nashville within 4-2, Duchene scored at 10:56 to make it 4-3, and Colton Sissons scored 34 seconds later to tie the game 4-4 at 11:30.
Kings coach Todd McLellan called a timeout after Sissons scored to allow them to regroup, and Iafallo, who hit the post with 4:15 remaining, scored the game-winner off the rebound of Brown's shot. It was his first goal of the season.
Brown scored into an empty net with 36 seconds left to make it 6-4, and Drew Doughty scored from the Los Angeles zone as Saros was skating to the bench to make it 7-4 with 29 seconds remaining.
"We want to be what we saw for the first two periods," McLellan said. "We're accomplishing things as a team. We're starting to develop our identity. We slip back a little bit every now and then, and that's why there is so much inconsistency in our game. But once we get it and everybody buys in for a full 60 (minutes), we'll be fine."
Kopitar gave the Kings a 2-1 lead at 11:14 of the first period when he scored on a one-timer off a pass from Brown.
Kopitar has two goals and five assists during a four-game point streak.
Kyle Clifford made it 3-1 at 6:53 of the second when he redirected a shot by defenseman Joakim Ryan, and Kovalchuk extended the lead to 4-1 at 18:12.
The Predators took a 1-0 lead at 4:55 of the first on Mikael Granlund's sharp-angled shot from the edge of the left face-off circle.
Amadio tied it 1-1 at 8:47 when he scored off a centering pass from Sean Walker that was tipped by Kovalchuk.
Nashville's Filip Forsberg failed to score a goal for the first time this season but had an assist on Granlund's goal to extend his point streak to five games (four goals, three assists).
