Despite trailing by 14 points at half-time, the Wolves fought back and outscored the Warriors 32-18 in the third quarter.
As the game headed into overtime, back-to-back three-pointers from Anthony Tolliver and Jerryd Bayless helped the Wolves take an early advantage, until Steph Curry tied the scores from the corner.
Towns was then fouled by Kevin Durant on the inbounds pass, resulting in the game-winning free throw.
— Timberwolves (@Timberwolves) March 30, 2019
Kyrie Irving also sealed victory with half a second to go as the Boston Celtics beat the Indiana Pacers 114-112.
His lay-up helped the Celtics move into fourth place in the Eastern Conference standings, and gave them the edge for home court advantage in a potential first-round series between the two teams in the post-season.
Irving led the way on 30 points and five assists while Al Horford added 19 points and Aron Baynes 13.
The Denver Nuggets completed a series sweep of the Oklahoma City Thunder with a 115-105 win.
Jamal Murray scored 27 points while four other players also finished in double figures.
Meanwhile, the Portland Trail Blazers won their season-high sixth game in a row, defeating the Atlanta Hawks 118-98.
The side had a 16-0 run in the third quarter to help them outlast the Hawks and keep them in third place in the Western Conference standings.
Elsewhere, Donovan Mitchell poured in 35 points as the Utah Jazz downed the Washington Wizards 128-124 and the Los Angeles Lakers ended the Charlotte Hornets' four-game winning streak with a 129-115 victory.