World number one Dustin Johnson finished in joint-second, one shot off the lead on nine under par in his return to action after injury.
The tournament had been set to go to a play-off with three players tied at the top before Harman's dramatic finish.
"It's a lot of emotion but this feels really good," the 30-year-old said.
Harman made a bogey on the 15th but birdies on the last two holes saw him claim victory with a four-under-par 68 on the final day.
It was his second win on the PGA Tour to go along with his 2014 John Deere Classic triumph.
Patrick Reed led by one shot overnight but eventually finished five shots off the lead in a tie for 12th place after struggling to a three-over-par 75.
Johnson set the clubhouse pace with his round of 67, the joint-lowest round of the day, which also saw him birdie 18.
He was then joined by Pat Perez, who was three under through four holes and recovered from a double bogey at 14 to birdie two of the final three holes for a 68.
Spain's Jon Rahm finished fourth, two shots off the lead, while England's Paul Casey was another to have a strong final round, his 68 helping him into a tie for 12th.
American Brian Harman made a dramatic 28-foot putt to birdie the final hole and win by one shot at the Wells Fargo Championship in North Carolina.