Alec Martinez and Wayne Simmonds also had goals as the Kings won the season series with Nashville 3-1. Brown sealed the victory with an empty-netter with 62 seconds left.
Jonathon Blum and J.P. Dumont each scored for Nashville.
The Predators came in trailing fifth-place Los Angeles by only three points in the tight Western Conference, but they snapped a two-game winning streak and missed out on a chance to move from ninth into one of the top eight spots.
Nashville outshot the Kings 32-18 but just couldn't beat Jonathan Bernier, who has been in net in all four games this season against the Predators.
Nashville took 10 of the game's first 11 shots and Bernier stopped every one, including back-to-back attempts by rookie Blake Geoffrion and Mike Fisher.
Los Angeles took advantage of a faceoff to goalie Pekka Rinne's left. Dustin Penner got the puck along the boards and passed it back to Kopitar, who wristed it past Rinne with 6:29 left in the first for the 1-0 lead.
That quieted the fans and not even a big check by Jordin Tootoo could stir them.
The Predators tied it off a faceoff in their offensive zone, and Blum scored his second this season on a slap shot at the blue line with 2:31 to go to revive the crowd. Captain Shea Weber did his part, too, in the final seconds, sending Kings left wing Alexei Ponikarovsky into the boards with a crushing hit.
Nashville turnovers helped Los Angeles score twice in the second period and take control of the game. A turnover by the Predators in their offensive zone started a 3-on-2 breakaway, and Brown found Martinez with a cross-ice pass that he wristed past Rinne at 2:16 of the second for a quick 2-1 lead.
Ponikarovsky collected the puck near the blue line, keeping it inside the Kings' zone and getting it to Kopitar, who passed it cross-ice. Simmonds took it and sent a one-timer from the left circle between Rinne's stick and the post with just under 8 minutes left in the period.
Nashville finally got the puck past Bernier with 7:17 to go in the game when Dumont, a healthy scratch the past two games, redirected Cody Franson's shot.
Brown's empty-netter finished off the win, and the Kings narrowly missed a second when the puck bounced off the right post just before the buzzer.
Notes: Kopitar set the franchise record by playing in his 325th consecutive game, topping the mark previously set by Marcel Dionne at 325 between 1978 and 1982. Kopitar's streak started March 21, 2007, in his rookie season. ... Kopitar now has points in five straight games (five goals, four assists). ... The Predators scratched forward Joel Ward with a lower-body injury. Ward had just matched his career high with three goals in the previous three games. ... Los Angeles improved to 33-7-2 when scoring three or more goals and 32-6-2 when allowing two or fewer goals.