It never came after what they believed was a questionable call.
Minnesota led by nine with 1:41 remaining in the extra period before Curry made three 3-pointers during an 11-2 run, including the tying one from the corner with 0.5 seconds left.
But Towns was fouled by Kevin Durant on the ensuing inbounds pass on a lob play to the rim that appeared to be thrown too high. He made the first free throw to give him 15 points, intentionally missed the second and time expired as Towns secured the offensive rebound.
''I'm just real proud of us, especially with the game the way it was going,'' Towns said. ''A team of that caliber, to be down so big and then say we're not going to quit.''
Curry made 11 3-pointers and scored 37 points for the Warriors, who dropped into a tie with Denver for the top seed in the Western Conference playoffs. Durant added 23 points.
''First of all, we're up 14 coming into the third, we can't put ourselves in that position,'' Durant said. ''But this is the NBA and teams make a run, especially at home. Momentum can shift very quickly. So, they made it a game. But I thought we deserved to play five more minutes.''
Andrew Wiggins scored 24 points, his fifth consecutive game with 20 points or more, for Minnesota. Josh Okogie scored 21 points and Jerryd Bayless scored 17 as eight Timberwolves scored in double figures. Gorgui Dieng had 14 points and 10 rebounds for his first double-double of the season.
The Minnesota bench outscored the Golden State bench 49-25.
''That's a big win,'' Timberwolves coach Ryan Saunders said. ''Especially when you're struggling, when you're down in numbers. In a lot of ways, that's a win that can propel you.''
Towns battled foul trouble and missed his first six shots from the field, while Wiggins fouled out with 2:21 left in overtime.
''It shows our fight,'' Wiggins said. ''We're never going to give up, no matter what.''
The Warriors, who had won two consecutive games, led by as many as 19 points in the first half before the Timberwolves rallied in the third quarter. Minnesota put together a 32-13 run to end the period.
Anthony Tolliver's 3-pointer to start the fourth quarter gave the Timberwolves their first lead since 16-15 in the first quarter. Minnesota led by as many as nine in the fourth quarter before Golden State tied the game at 115 on Draymond Green's free throw with 21 seconds remaining. Wiggins had two chances to win the game in regulation.
Tolliver, Bayless and Okogie each hit 3-pointers in overtime to help Minnesota build its nine-point advantage. Curry's tying, turnaround 3-pointer came immediately after Durant appeared to hit a tying 3. But Keita Bates-Diop was called for a foul on Durant before the shot. The Warriors were unhappy with that call and the ensuing call on Durant.
''Down the stretch, it's just tough an effort like that gets sabotaged, for sure,'' Curry said. ''Went back and looked at those calls over and over and over again, and still don't understand either one of them.''
Klay Thompson scored 20 points for the Warriors, who dropped to 0-5 in overtime games this season. Golden State was held to 18 points and had zero assists in the third quarter after having 20 assists on its 26 field goals in the first half.
''Everybody's feeling great at halftime, and the third quarter was a disaster,'' Warriors coach Steve Kerr said. ''We stopped playing. I give Minnesota a lot of credit. They competed. They played really hard and got back in the game. We stepped on our own toes in that quarter.''
Warriors: C DeMarcus Cousins had 16 points and nine rebounds before fouling out with 7:06 to play. ... F Jordan Bell was active after missing Wednesday's win in Memphis because of a team suspension. ... Curry, Cousins and a group of approximately 20 that included coaches and staff attended the Minnesota Twins' season opener on Thursday.
Timberwolves: Friday was Minnesota's only win in the four-game season series with Golden State. F Taj Gibson (strained left calf) and F Luol Deng (left Achilles tendon soreness) did not play. Gibson has missed four consecutive games, while Deng has missed 14 straight.
GUEST OF HONOR
NBA Commissioner Adam Silver was in attendance Friday. Saunders fondly recalled Silver reaching out to him after he was named interim coach Jan. 6.
''I'm looking forward to seeing him,'' Saunders said, ''because he's done great things for our league and continues to do great things.''
Warriors: Host Charlotte on Sunday.
Timberwolves: Host Philadelphia on Saturday.