Adam Fox and K'Andre Miller each had an assist on Chytil's goal after each defenseman scored in the second period.
The Rangers (23-12-7) were down 2-0 in the first period and 3-2 early in the third, but still extended their point streak to six games (4-0-2). They are 12-2-2 since Dec. 5.
"It's pretty important for a team like ours, especially if we make the playoffs," Panarin said of the comeback win. "It's an important skill if we can go together and keep it, it doesn't matter how things are going, we can do the same thing."
Jon Merrill, Kaprizov and Zuccarello scored for the Wild (22-14-4), who have lost three straight games (0-1-2). Sam Steel had two assists, and Fleury made 40 saves.
"We got some points, but we all know it's not good enough," Fleury said. "The standings are so close. To make playoffs you need points, we need to get wins. It's about getting out of this slump as quickly as possible to start winning again."
Merrill gave Minnesota a 1-0 lead at 12:36 of the first period with a shot through a screen from Joel Eriksson Ek beat Shesterkin on the far side.
The Wild took a 2-0 lead on Kaprizov's goal off the rush at 17:43.
"We were down 2-0 in the first, I thought we could have been winning after the first period," Fox said. "But we battled. If we play like that I think we're going to win a lot of games."
Miller scored on a breakaway at 2:37 of the second period to make it 2-1. Jimmy Vesey forced Matt Boldy into a turnover just inside the blue line. Miller won a foot race to the puck, fending off Boldy and beating Merrill, to go in alone on Fleury.