Kirill Kaprizov scored twice, and Gustavsson made 23 saves for the Wild (30-21-5), who have won two straight after losing five of their previous six (1-4-1).
"Really happy we got the win, but ultimately it was too close," Kaprizov said. "We need to find a way to win clean, not make it a chaotic end like that. Whether it be on the power play, get the goal when we need to, but yeah, good win."
Tommy Novak had a goal and two assists, Roman Josi had a goal and an assist, and Saros made 31 saves for the Predators (26-22-6), who were coming off a 7-3 win against the Florida Panthers on Saturday.
Duhaime scored nine seconds into the second to give Minnesota a 1-0 lead when Frederick Gaudreau's shot deflected in off him in front.
Kaprizov extended the lead to 2-0 at 1:37, finishing a rebound in the slot for a power-play goal, but Novak cut it to 2-1 at 17:03.
"Sometimes you just go through ebbs and flows, shots going in," said Novak, who has three goals in the past two games. "My linemates have been playing really good with me the last two games. Got a bounce to get a 2-on-1 today, and ended up going in. I think just trying to play hard every shift and trying to keep consistently high level of play."