The 20-year-old Californian rolled in a 26-foot putt from the fringe for a one-stroke victory over Bryson DeChambeau and Collin Morikawa in Blaine, Minnesota.
“I just proved to myself that I can be out here,” Wolff, the 2019 U.S. collegiate champion, told CBS television.
Five-time tour winner DeChambeau had eagled the same hole minutes earlier with a six-foot putt to temporarily take the lead by one stroke.
But Wolff was not done as he sank the winning shot for a six-under 65 to win in only his third tour start as a professional at 21-under 263.
“I was shaking, to be honest,” Wolff said of his emotions before the shot. “A lot of my putts weren’t going in today, and I was just kind of telling myself one’s bound to drop in.
“It happened at the right time. Just gave it a chance and it went over the right spot that I pointed out. And as soon as it dropped, the tears started coming. It’s a really special feeling.”
Fellow Californian Morikawa then missed a 22-foot putt that would have forced a playoff at TPC Twin Cities.
Morikawa, who is 24, and the 25-year-old DeChambeau both shot 66.
With the victory, Wolff is exempt on the PGA Tour through the 2021 season and will play in The Players, the Masters and the PGA Championship next year.
“I’ve changed forever, I guess,” he said.
Spieth was 19 years, 11 months and 17 days old when he won the 2013 John Deere Classic.
Canadian Adam Hadwin shot 67 to claim fourth at 18 under with Mexico’s Carlos Ortiz (64) and Wyndham Clark (68) a stroke back.
As many as six players were tied for the top on a warm afternoon ideal for scoring.
Wolff, who shared the overnight lead with Morikawa and DeChambeau, was the first to reach 19-under with a birdie at the 15th hole and Morikawa joined him a hole later.
DeChambeau pulled within one with his birdie at the 16th.
After all three took pars at the 17th, the fireworks began at the 18th with DeChambeau pushing his second shot to within six feet of the hole and the eagle that temporarily put him in front.
He ended up with five birdies, two bogeys and the eagle.
“Sometimes you’re just not going to get your way, you know?” DeChambeau said.
“Unfortunately the par-fives weren’t nice to me this week, except on 18 today. That’s just something that I’ve got to work on.”
Wolff, who has often played with fellow Californian Morikawa, finished with the winning eagle, five birdies and a lone bogey at the ninth hole.
Morikawa started the day with bogeys at his second and third holes before collecting his first birdie at the seventh and then made six birdies in eight holes on the back nine.
Reporting by Gene Cherry in Raleigh, North Carolina; Editing by Ken Ferris/Greg Stutchbury