Vlasic scored a power-play goal at 9:01 of the first period to put the Blackhawks ahead 1-0. He took a drop pass from Teuvo Teravainen and scored on a slap shot from the high slot with Ryan Donato and Tyler Bertuzzi providing screens in front of the net.
Gabriel Vilardi tied it 1-1 at 13:31, when he tipped in Logan Stanley's shot from the left point.
Martinez's first goal of the season gave Chicago a 2-1 lead at 19:47. Ilya Mikheyev passed from the right point across to Martinez, who skated into the left face-off circle and put a wrist shot over Hellebuyck's left shoulder.
Nino Niederreiter tied the game 2-2 at 13:10 of the second period. He skated along the goal line on the left side and lifted a backhander over Soderblom's right shoulder.
"Yeah, definitely it was huge, obviously," Niederreiter said of his goal. "I mean, we had a couple of looks, especially early on, which I obviously wish I could have put that one in already. But I knew I just got to stay with it, and it's going to come and go to the dirty areas, I guess. Very fortunate I was able to finish that."
Scheifele scored on a rebound of Connor's shot to put the Jets ahead 4-3 in the third, and Appleton scored an empty-net goal at 18:19 for the 4-2 final.
Winnipeg coach Scott Arniel said his team was aware that the Blackhawks could have benefited from the coaching change.
"It always happens when there's a change," he said. "The players are looking for maybe a second opportunity or more ice, different roles. There's also the guilt that they're a part of it especially a good coach like Luke getting let go.
"But honestly, we didn't look there. We just had to focus on our game, and we're not worried too much about what's happening across the hall. We have to make sure we play Jet hockey, and I thought we did a good job with that tonight."
Chicago, which has been outscored 21-10 during its losing streak, begins a three-game road trip at the New York Rangers on Monday.
"We obviously understand the urgency," Blackhawks captain Nick Foligno said. "Maybe it's good that we get on the road and get to know Anders a little bit better by being around him and being around each other. Yeah, we're going to have to keep working at this. It's not just going to turn around in one day. Obviously we'd like it to. It would have been nice to get him that first win. But we're going to find a way to get it in New York."
NOTES: Sorensen became the first Sweden-born coach in NHL history. … The 14 shots on goal were the fewest of the season for the Blackhawks. … Martinez left the game at 1:15 of the third period after a shot by Adam Lowry deflected off his stick and hit him on the right side of the face. Sorensen said he was being evaluated and "might miss some time." … Josh Morrissey played in his 610th game with the Jets/Atlanta Thrashers, passing Dustin Byfuglien for sole possession of second place among defenseman in franchise history. Toby Enstrom is first with 719.
