Johnson forced the playoff by sinking a sharply-breaking downhill 17-foot par putt at the final regulation hole, his ball catching the right edge of the cup and toppling in at the par-four 18th.
He overcame a five-stroke deficit over the final 13 holes for his first win since he injured his back falling down stairs on the eve of the U.S. Masters in April.
After a quiet summer, Johnson stamped his authority with a closing 66 for 13-under-par 267 in the first of the PGA Tour's four FedExCup playoff events.
"I feel the game’s finally back in form like it was leading into the Masters," Johnson said in a greenside interview. "I’m swinging everything really well, got a lot of control over the golf ball and I’m feeling really good.
"Getting a win here today gives me a lot of confidence going into next week and the rest of the playoffs."
Johnson, who won three consecutive starts before hurting his back, joins Justin Thomas as a four-times winner on the PGA Tour this season.
He has 16 career victories, and this was his first in a playoff.
But he seemed destined for second place after leaving himself with a difficult par putt at the 72nd hole.
With Spieth sitting within tap-in range for a certain par, Johnson had to hole the putt, and he read the break and the speed to near perfection to dish out a blow to Spieth.
"I felt I played really good today," Johnson said.
"The front nine I couldn’t hole a putt, burning the edges. Finally rolled in a couple of putts on nine and 10."
Spieth carded 69. He and Johnson finished four shots clear of Jon Rahm and Jhonattan Vegas.
Spieth, who had earned a reputation as a closer having converted nine of his previous 10 54-hole leads into victories, appeared on the verge of a runaway win after making two birdies in the first five holes to jump five clear of Johnson.
But Spieth's eight-iron at the par-three sixth failed to clear the water hazard guarding the green, and the subsequent double-bogey left him only three shots clear of Johnson.
Four holes later they were tied, and there was never more than a shot separating them from that point.
(Reporting by Andrew Both in Cary, North Carolina, editing by Ed Osmond and Frank Pingue)
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