Seth Jones had two assists, and Jaxson Stauber made 29 saves for the Blackhawks (20-32-5), who have won four straight.
"I've said it all year, these guys don't quit," Chicago coach Luke Richardson. "We got some penalties against us, and then they scored on the power play. We were down in the game and we're tired. It's easy to fold the tent, pack it in, but these guys, they are resilient.
"I think they like playing with each other. Once we got a little sniff and we got that first goal, I think they started to believe again and started working for each other."
Jamie Benn had a goal and an assist, and Jake Oettinger made 19 saves for the Stars (30-16-12), who blew a three-goal lead and lost their fifth in a row (0-3-2) and eighth in their past 10 (2-3-5).
Tyler Seguin nearly tied it at the end of the third period, but his shot did not go in until after time expired. Radek Faksa almost scored for Dallas with 1:40 remaining, but video review determined the puck did not completely cross the goal line.
"It's definitely a punch in the gut," Stars forward Jason Robertson said. "Every team is good in this league, so not taking any credit [away] from Chicago. One of those games where you make those little mistakes going east-west and gave up three 2-on-1 goals today, and that can't happen, especially when we're in this little rut to begin with.
"It's unacceptable and I'm sure everyone in here is going to regroup and get back to what (Dallas coach) Pete [DeBoer] wants us to do."
Kane scored to make it 3-2 at 15:24 of the second period after a giveaway in the neutral zone by Seguin, then tied it 3-3 at 18:05 on a 2-on-1 with Philipp Kurashev.
"Where do you start? You build a 3-0 lead, just got to be a smarter team," DeBoer said. "I think we are a smarter team than that, but obviously weren't tonight. Their power-play goal [to make it 3-1] gives them a little momentum, and you've got to be smart enough to kind of stop the bleeding there. Instead, we compound it with some turnovers and give them life that you don't need to give them."