Nyqvist gave the Wild a 1-0 lead with a power-play goal at 9:00 of the first period. Mats Zuccarello passed to Johansson in the slot, who fed Nyqvist at the side of the goal for a tip-in.
"I feel like I've had my chances in previous games too but haven't been able to put the puck in the net so obviously that was nice to do tonight," Nyqvist said.
Braden Schneider tied the game 1-1 at 12:45 scoring from the left circle on a rebound of a shot by Urho Vaakanainen.
Faber gave the Wild a 2-1 lead at 15:00 with a shot that deflected off Rangers forward Gabe Perreault's stick and went past Shesterkin.
The Rangers trailed after the first period despite outshooting the Wild 18-7 and having a 28-19 advantage in shot attempts.
"Overall, we came out on the attack," Rangers coach Peter Laviolette said. "... We needed a win, and guys found a way to do that, so it was good to get a 'W' in a game that went up and down and kind of went back and forth."
Miller tied it 2-2 at 2:51 of the second period, after the puck bounced off the end boards to him in the left circle and his shot went over Gustavsson's right shoulder.
Chris Kreider gave the Rangers a 3-2 lead at 7:21 off a giveaway in front of the net by Wild defenseman Declan Chisholm.
"I feel like we, other than a few shifts, I felt like they had some O-zone time on us," Johansson said. "I think we played all right. We created enough to win. I feel like a couple of bad breaks on some of those goals, but what are you going to do?"