Owners of the sixth-worst record in the NBA, the Timberwolves erased a 17-point deficit to force overtime.
Wiggins, who finished with 32, drove around lockdown defenders Andre Iguodala and Draymond Green to knot things at 106 in regulation.
KAT finished with 20 points, 12 rebounds, 4 assists, and 2 blocks. He did some real damage in the final period.
Reserve swingman Shabazz Muhammad led all scorers with 35 points.
But the key to victory was their defense on reigning MVP Stephen Curry. Typically when the Warriors force opponents to switch their center onto the star point guard, the possession ends with an easy three points.
Fortunately for Minnestoa, Towns is anything but your typical center. The Timberwolves welcomed such switches, and it worked.
Particularly in overtime, when the rookie big man forced Curry into an ugly attempt.
Steph shot just 7-of-25 in the game for 21 points and didn't make a field goal in the first half. He dished out 15 assists.
"It wasn't even about being a part of history," Towns said after the win. "You want to go out there and win. Every game, you want to go out there and you want to be a very competitive team that goes out there and wins."
Now 69-9, the Warriors need to win their last four games to break the 1996 Chicago Bulls record 72-win season.
After the loss, Golden State seemed tired of chasing the record.
"It's human nature to where, all right, kind of ready for the regular season to end," Green said. "Talking 82 games, we get bored with that after awhile. And that's no excuse, just, I'm always give it to y'all real, and that's about as real as I can be. It's kind of at a point now where you're ready for the regular season to be over."
They'll play the San Antonio Spurs and Memphis Grizzles each at home and on the road to close out the season. The Spurs haven't lost at home in 54 straight.