Hutton, a 27-year-old making just his second NHL start after spending most of his career in the AHL, picked up his first career win. Nashville coach Barry Trotz chose to rest starter Pekka Rinne after the team's Saturday night win in Montreal.
The Predators (5-3-1) recorded only seven shots in the second period, but made three of them count as they took control of the game. The win extended Nashville's point streak to five games.
Nystrom opened the scoring 20 seconds into the second period with a shorthanded effort. The left-winger blocked a Dustin Byfuglien point shot, picked up the puck in the neutral zone and went the length of the ice before beating Jets goalie Ondrej Pavelec with a wrist shot for his second goal of the season.
Hornqvist's third of the year made it 2-0 when he intercepted a Pavelec clearing attempt in the corner, sent the puck toward the crease, and banked it into the empty net off the skate of Jets defenceman Zach Bogosian.
Cullen added to the lead with a high shot from the left faceoff circle at 14:57.
Pavelec stopped 23 of the 26 shots he faced.
The Jets (4-5-0) ruined Hutton's shutout bid at 9:13 of the third period when Kane jammed in a loose puck for his fifth goal of the season.
Cullen had the best chance in a scoreless first period when he came out of the penalty box and scooped the puck up for a breakaway, only to be stopped by Pavelec. The shots were 15-12 Jets after one period.
The Jets wrap up their season-long six-game homestand on Tuesday night against Washington. The Predators visit Minnesota on Tuesday before heading home for a rematch against Winnipeg on Thursday.
Notes: Rookie defenceman Seth Jones logged more than 26 minutes of ice time for the Predators. … Forward Devin Setoguchi was back in the Winnipeg lineup after being a healthy scratch against St. Louis on Sunday night. … The game marked the first meeting between the franchises as Central Division rivals.