Spieth, a Dallas native who doesn't turn 20 for another two weeks, hit a short par putt to earn a spot in the field at Muirfield.
He is the first teenager to win since Ralph Guldahl took the Santa Monica Open in 1931.
"I didn't think it would happen this early," said Spieth, who turned pro in December after an All-American season as a freshman at the University of Texas. "I had a plan. I guess the plan got exceeded."
Spieth started the day six shots behind third-round leader Daniel Summerhays. A bogey on the first hole left him seven back with 17 to play.
But Spieth forced his way into the playoff with three straight birdies. The last came when he holed out of the bunker from 44 feet. Spieth caught a break when that 44-footer took a pair of fortuitous bounces.
"The shot on 18 was the luckiest shot I ever hit in my life," Spieth said. "The fact that it bounced right and hit the pin and dropped down to the cup, it's just extremely fortunate."
Spieth, Hearn and Johnson all made par on the first four playoff holes. Spieth simply made one more to stave off Johnson and Hearn on the fifth.
Summerhays, a former BYU golfer, held a two-shot lead entering the final round, but struggled with four straight bogeys on the front nine. He fought his way back into contention with a 34 on the back nine, but bogeyed the par-4 18th hole to finish one shot out of the playoff.
U.S. Senior Open
In Omaha, Neb., Kenny Perry completed his comeback from 10 shots down after the second round to win the U.S. Senior Open. Perry shot a final-round 63 at the Omaha Country Club and won his second straight senior major by five strokes over Fred Funk. He set two tournament records over the last 36 holes for the 64-63 finish and for the size of deficit he overcame.
In Inverness, Scotland, Phil Mickelson birdied the first playoff hole to beat Branden Grace and win the Scottish Open, earning his first victory in Europe in 20 years.
In Waterloo, Ontario, Hee Young Park beat Angela Stanford on the third playoff hole to win the Manulife Financial LPGA Classic.
John Deere Classic
* - Jordan Spieth -19
Zach Johnson -19
David Hearn -19
Martin Flores -18
Jerry Kelly -18
Daniel Summerhays -18
* - wins playoff