Finnish midfielder Robin Lod tapped home from Niko Hansen's deflected cutback for the 81st-minute winner.
The Sounders had enjoyed a spectacular start to the season, but have now only managed one win from their past four games.
Seattle remains top of the Western Conference with 29 points from 14 games, but second-placed Sporting KC are three points behind with a game in hand after winning three in a row. Minnesota moves up to sixth with the win.
United had two good first-half chances from set pieces, with Emanuel Reynoso's free-kick going marginally off target and Bakaye Dibassy's flick from a corner going wide too.
MLS equal top scorer Raul Ruidiaz was denied by United keeper Tyler Miller too, while the Sounders thought they had the lead in the 76th minute. Xavier Arreaga headed home Joao Paulo's free-kick but it was disallowed for offside.
Lod had a penalty shout turned down before a minute later finding space inside the box to finish from Hansen's pass.