Marcus Foligno pulled the Wild into a 2-2 tie at 14:40 of the third period when he controlled his own rebound while falling over Dallas goalie Jake Oettinger at the left post and backhanding the puck into the net.
"There's a lot of belief in here," Foligno said. "These are two unreal teams, and we've got to understand they're a heck of a hockey club and it's going to take this type of effort every night against this squad. There's no pouting in here. We took a lot of positives out of (a 4-3 double overtime loss in Game 3 on Wednesday) and we got rewarded tonight."
Brock Faber had a goal and an assist, Kaprizov had two assists, and Jesper Wallstedt made 43 saves for the Wild, the No. 3 seed in the Central Division.
"We just keep fighting, keep getting pucks to the net," Wallstedt, who made nine saves in overtime, said. "We keep playing and we never give up. It felt like we really deserved this one."
Matt Duchene and Mikko Rantanen each had two assists, and Oettinger made 40 saves for Dallas, the No. 2 seed from the Central Division.
"These are player games," Dallas coach Glen Gulutzan said. "We're making tweaks every game, tweaks everyday. We've got a couple things in our mind already for (Sunday) that we can button up for the next game, but it's the war of the blue paint. You need to get in there and find some pucks, get some sticks on things and win some battles when it's that contested.
"It's really now hand-to-hand combat, and blue paint (battles), for both sides."
The best-of-7 series is tied 2-2. Game 5 will be at American Airlines Center in Dallas on Tuesday (8 p.m. ET; FDSNWI, FDSNNO, Victory+, ESPN2, TVAS2, SN360).
"We got home ice now and it's best-of-3," Rantanen said. "This group has been in this situation before, so we're pretty familiar."
Jason Robertson gave the Stars a 1-0 lead with a power-play goal on a snap shot from the bottom of the left circle at 5:35 of the first period. Duchene had the puck in the slot and passed to Robertson in stride toward the net in the left circle.
Faber made it 1-1 when his wrist shot from the right point deflected off Stars defenseman Miro Heiskanen's left leg in the slot and past Oettinger at 13:52.
"I just think our mindset is more taking whatever the game is giving us. Coming out, and playing and staying with it," Minnesota coach John Hynes said. "I think we've done a good job of that and I think there were some lessons out of the first couple of games of making sure that we're just playing. We're going over the boards and regardless of what the situation is or what happens, we're focused on playing how we need to play regardless of the circumstances."