He carded 68 to finish at 10-under-par 278 for his second PGA Tour victory, one stroke ahead of fellow Americans Johnson (67) and Pat Perez (68).
Spaniard Jon Rahm (71) finished two strokes behind, while overnight leader Patrick Reed (75) struggled with his driver and faded to end five strokes behind.
World number one Johnson, playing for the first time since falling down stairs on the eve of the Masters a month ago, was seeking to become the fifth player in PGA Tour history to win at least four consecutive starts.
After sinking a 15-foot birdie at the par-five 18th, he looked set to be part of a playoff until Harman birdied the final two holes, a five-footer at the 17th setting up his final-hole heroics.
Harman said it would be wrong to call his winning putt unbelievable.
“I could believe it because I’ve been rolling it so well this week but when that thing was about a foot short I said, 'That thing’s going home’,” Harman, 30, said in a greenside interview after collecting his second PGA Tour victory.
“I’ve been working really hard... just trusting that I’m pretty good at what I do. This feels really good. It’s a lot of emotion for sure.”
Harman expected his 2014 John Deere Classic victory to open the floodgates, but instead spun his wheels until recently.
“After I won the first one I thought I was going to be there a lot and it didn’t work out that way,” he said.
“I struggled over the next two years, never really got into the hunt. I started getting into the hunt at the beginning of this year and started feeling like it was coming around, so I’m glad to get it validated.”
Despite not winning, Johnson, 32, was happy with the state of his game in his first tournament after five weeks off.
“I didn’t know what to expect this week,” he said. “The first couple of days I didn’t play that great but really played nicely on Saturday and Sunday so I’m happy with where the game is going into next week."
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ken Ferris)
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