Patric Hornqvist also scored and had two assists and Matt Cullen added two assists as Nashville snapped a six-game losing streak to the Ducks.
"We got some good bounces around the net and that was key for us," Hornqvist said. "It's been a bumpy road for us but we're not out of it yet. We have to fight to the end."
Nashville, 10th in the Western Conference, trails eighth-place Phoenix by seven points with five games to play.
Hornqvist has 13 points (six goals and seven assists) in his last 10 games.
Pekka Rinne stopped 27 shots for the Predators, who led 4-1 after the first period. Rinne tied former Predators goalie Tomas Vokoun for most franchise wins with 161.
"It feels great to tie Tomas for most wins; he's had a very successful career," Rinne said. "I've had up and downs this season but it's good to see the crew get the win. We're going to battle to the end."
Wilson jammed the puck past Ducks goalie Jonas Hiller for his ninth goal of the season at 5:36 of the first to give the Predators a 1-0 lead.
Wilson added his second of the game and 10th of the season when he took advantage of an Anaheim miscue in front of the Ducks' net and his backhander made it 2-0 with 7:19 left in the period.
"It's good to snap our losing streak against them," Wilson said. "We knew they are never out of a game. We're not out of the playoffs yet. We're looking forward to going into the `Shark Tank' tomorrow and it will be fun against San Jose."
Just 1:11 later, Fisher scooped up a loose puck behind the Ducks' net and wrapped it around into an open net past Hiller to make it 3-0. It was Fisher's 19th goal of the season.
Palmieri got Anaheim on the board with his 13th of the season when he beat Rinne with a wrist shot with 5:00 to go in the first.
Hornqvist scored his 18th goal as he drilled it past Hiller 2:05 later to increase Nashville's lead to 4-1. Hiller, who was playing for the first time in three games, was immediately replaced by rookie Frederik Andersen following Hornqvst's goal.
Andersen left the game after the second period after suffering an upper body injury. Hiller returned to the net for the third period.
The Ducks cut the lead to 4-2 following Winnik's fifth goal at 3:18 of the second, but Fisher got his second goal of the game with 4:00 left in the period.
"They (Anaheim) are something like plus-36 in the second period, so I was concerned going into the second period," Nashville coach Barry Trotz said. "Our fifth goal was huge for us. After that, I didn't feel any pressure in the third period."
The Ducks, who outshot the Predators 24-16 in the first two periods, had two goals nullified late in the second.
"We played a solid defensive game," Trotz said. "We got a couple early goals on Hiller and that helped us. There's no quit in these guys. They are very professional about their jobs and we've been playing sharper lately."
Despite the loss, Anaheim remains atop of the Pacific Division with 108 points, one point ahead of the second-place San Jose with both teams having four games remaining.
The Ducks, who completed its three-game homestand on Friday, also trail the Western Conference-leading St. Louis Blues by three points for the overall points lead.
Anaheim coach bruce Boudreau said his team has to start playing harder earlier in the game.
"We can't let this happen," he said. "We have to find a way. Sometimes, for one reason or another, we're not getting the job done early. We're a really good team with the lead, but we just haven't had the lead lately. We just didn't answer the bell tonight."
Nashville, playing for the first time in five days, climbed out of last place in the Central Division, moving past Winnipeg.
The Predators prevented the Ducks from tying a franchise record of 28 home wins and 110 points in a season.
Friday was the first time the Ducks hosted Nashville since Feb 16, 2013. ... Trotz is the only coach in the Predators' 15-year history and is the longest tenured coach in the NHL (1,190 games) and has the second-longest tenure in all four major sports behind the San Antonio Spurs coach Greg Popovich. ... The Predators are the only team in the league not to give up a short-handed goal this season but have also scored the fewest short-handed goals (2) in the league.
Copyright by STATS LLC and The Associated Press
ANAHEIM, CA - APRIL 4: Goalie Jonas Hiller #1 of the Anaheim Ducks prepares his crease to play the Nashville...