With their first win in the best-of-seven series, the Timberwolves are hoping to become the first team in NBA history to come back from 3-nil down and make it to the finals.
The win came on the back of Anthony Edwards' game-high 29 points and Karl-Anthony Towns' recalibrated long-range shooting form.
Winning the series was not on the team's radar in Game 4, Edwards insisted afterward.
"We just wanted to get one game and extend the series," he said.
"I've never been swept in my career. I didn't want to get swept. Not on their home court, hearing their fans talk trash all day."
Luka Doncic posted a 28-point, 15-rebound, 10-assist triple-double for the fifth-seeded Mavericks, who won the first two games of the series in a previous trip to Minneapolis.
After Minnesota led by as many as 12 early on, the teams battled on even terms for the first 18-plus minutes of the second-half, with a Towns 3-pointer with 5:41 remaining giving the Timberwolves a 92-90 lead.
The visitors never trailed again, thanks in large part to Towns, whose 3-pointer on Minnesota's next possession made it a five-point game.
Dallas' best chance to draw even down the stretch came when Doncic misfired on a 3-pointer with the hosts trailing 95-92 with 3:18 to go.
When Towns connected again from beyond the arc with 2:54 to go, and Edwards followed with a bank shot in the lane 67 seconds later, the Timberwolves had broken things open at 100-92.
"That was on me," shouldered Doncic. " didn't give enough energy."
A desperation 3-pointer by Doncic as he was being fouled with 13.2 seconds left kept the Mavericks alive at 103-100, but Doncic missed the subsequent free throw.
After a timeout, Naz Reid got behind the Dallas defence for a game-clinching layup with 11.3 seconds remaining.
"We'd never lost three in a row until earlier in the playoffs. We've never lost four in a row," Timberwolves coach Chris Finch said.
"The guys responded. Fun team to coach because they always believe they can win no matter what."
"They won one game," Doncic said. "We got to focus on the next one."
-Reuters