Josi and forward Philip Tomasino each had a goal and two assists, and Matt Duchene had a goal and an assist for Nashville (40-25-4), which has won three of four and moved four points back of the St. Louis Blues for third in the Central Division.
The Predators also remained one point ahead of the Dallas Stars, who hold the second wild card from the Western Conference.
"We thrive on hard, fast, competitive hockey," Duchene said. "We thrive on it, meaningful hockey. We're a team that, earlier in the year, some of the top teams were beating us. We addressed it, talked about it. Since then, it's been the ability to get our game up to the top level. That's us in a nutshell tonight."
Mats Zuccarello and Kirill Kaprizov each had a goal and an assist, and Marc-Andre Fleury made 27 saves for Minnesota (43-21-5), which had gone 9-0-1 in its past 10 games since losing 6-2 to the Predators on March 13. The Wild are in second place in the Central Division, three points ahead of the Blues.
Josi made it 1-0 with a power-play goal at 5:10 of the first period, beating Fleury with a wrist shot glove side.
"We have to find a way to stay the [heck] out of the box because, obviously, they've got special players and special power-play players and they hurt us," Minnesota coach Dean Evason said. "Our 5-on-5 game is great. [Saros] is fantastic, a fantastic goaltender. They blocked a lot of shots. I don't know how many attempts we had at the net. We did a lot of good things. We're going to take some serious positivity out of this game, but disappointed."