Bryan Little added a goal and an assist.
Dustin Byfuglien and Blake Wheeler scored for Winnipeg.
Connor Hellebuyck stopped 31 shots for the Jets, who moved to 46-19-10.
Ryan Ellis and Viktor Ardvidsson each had a goal and an assist for the Predators.
Roman Josi and Kyle Turris had singles for Nashville.
Juuse Saros turned aside 43 shots to post the win for the Predators, who dropped to 48-17-10 on the season.
Roman Josi scored at 10:09, one timing a Ryan Ellis feed past Jets goaltender Connor Hellebuyck beating him high blocker side from the top of the right faceoff circle to give Nashville a 1-0 lead.
The goal was his 11th of the season.
Kyle Turris scored on the power play with less than a second left in the first period after the Predators kept the Jets pinned in their own end.
Mattias Ekholm snagged a clearing attempt by Andrew Copp before sliding the disc to Turris who beat Hellebuyck just inside the left post with .3 of a second left in the period to put the Predators up 2-0.
The goal was his 14th of the season.
With P.K. Subban off for hooking the Jets power play went to work as Kyle Connor snapped up a rebound off a Bryan Little shot and lifted the puck past Nashville goaltender Juuse Saros to pull Winnipeg within a goal 3:49 into the second period. The goal was his 4th in his last 3 games and 7th tally in his last six contests.
Ryan Ellis restored the Predators two-goal cushion at 6:43 of the second, one timing a Kevin Fiala feed off a face off win in the Jets end to make it 3-1 Nashville.
The goal was his 8th of the season.
Andrew Copp appeared to have scored Winnipeg's second goal of the game at 8:59 of the second period but was ruled no goal due to a distinct kicking motion.
The call was confirmed by video replay.
Dustin Byfuglien pulled the Jets to within a goal at 11:29 with a power play marker off a shot from the right point that found it's way through Saros's pads to make it 3-2.
Bryan Little tied the game at 13:53, taking a Kyle Connor feed in Nashville's end before moving in to the top of the right faceoff circle and firing a slapshot past Saros, beating him blocker side with a high shot.
The goal was his 15th of the season.
Viktor Ardvidsson made it 4-3 Nashville at 15:18, shovelling a loose puck past the goal line during a goal mouth scramble to make it 4-3 Nashville.
Blake Wheeler tied the game just under a minute into the third period, grabbing a rebound off a Joe Morrow shot before skating in and sliding the puck between Saros's pads to make it 4-4.
The goal was his 20th of the season.
The game went to overtime.
There was no scoring in overtime and the game went to a shootout.
Nashville-Ellis (miss), Fiala (miss), Forsberg (miss)
Winnipeg-Little (miss), Schiefele (score), Laine (miss)
The Jets close out a six-game home stand on Tuesday against Boston while Nashville returns home to host Minnesota.
SHOTS-NASHVILLE 36 WINNIPEG 47