Having fallen behind to an Angelo Rodriguez strike at Audi Field, United responded in the final half hour.
Ulises Segura drew the home side level before substitute Darren Mattocks scored the winner, extending the club's unbeaten run to three matches.
The result moved Wayne Rooney and his team to within two points of the sixth-placed Montreal Impact in the Eastern Conference, while Minnesota sits ninth in the west.
While United still has seven matches left in the season, the win is a major boost to the club's playoff hopes as a tough closing run to the season looms large.
Next up is a clash with the New York Red Bulls, who have already clinched their place in the playoffs, before a showdown with the Impact on Sept. 29 that could prove decisive in the Eastern Conference race.
While Minnesota could, in theory, still make a playoff run, the loss leaves the club 12 points back of the sixth-placed Seattle Sounders, who also hold a game in hand on Adrian Heath's side.