The Wild, who had won the last two games heading into the night's potentially series-ending clash, were trying to make the second round for the first time in 11 years.
They were the first to get on the board six minutes and change in. Quinn Hughes was the first scorer as he was played in by Marcus Foligno before burying his shot.
That turned out to be the only goal of the opening frame, where Minnesota outshot Dallas 7-4. Neither team had a power play in the first 20 minutes, with the intensity settling into the Grand Casino Arena.
The Stars responded in the second with two goals to take what would be their only lead of the night. First, Wyatt Johnston made the most of their man advantage as Yakov Trenin was sent to the penalty box for interference. The centre was afforded enough space in a few feet from the goal and finished off Mikko Rantanen's centering pass.
Later on in the second, the guests went in front. After succesfully killing another power play, Mavrik Bourque got in on the scoring as he was the first to get his stick on a loose puck in the goal crease.
Behind for the first time in the game, the Wild responded not too long after. After failing to properly clear their lines, the Stars gave the puck away to Hughes, whose effort was tipped aside to Vladimir Tarasenko, who converted his chance while falling to the floor. The marker was Tarasenko's 50th NHL playoff goal.
With the score 2-2 heading into the third, the game was poised for a thrilling end as the Stars looked to keep their season alive.
Both teams were able to kill a 40-second power play each while having 80 seconds of four-on-four hockey. Just after the midway mark of the final period, Hughes scored his second of the night. Going towards goal on the left wing, the shot from the defenceman beat Jake Oettinger to give the Wild the go-ahead goal.
Dallas spent the final nine and a half minutes trying to find the levelling goal to potentially force overtime. Yet, none of their 10 efforts in the period could beat Jesper Wallstedt.
The Stars threw on an extra attacker towards the end, hoping that would help them in finding the back of the net. Unfortunately for them, they could not make the most of it, with Matthew Boldy closing out the game with a pair of empty net goals.
The Wild move on with a 4-2 series win and will face the President's Trophy winners Colorado Avalanche in the second round.