Henley's six-shot overnight lead was down to four when Scottie Scheffler birdied the 18th to complete a flawless closing 62 and the gap was three when Henley dropped his first shot of the week on the fifth.
However, the 33-year-old promptly birdied the next three holes and cruised to a fourth PGA Tour title, his first since the 2017 Houston Open.
Henley, who had failed to convert any of his last five 54-hole leads into a victory, told Golf Channel: "I just tried to learn from my past and my screw-ups.
"All of those events I didn't close, they hurt and you don't know if you'll ever get to win another one, it's so hard out here. To come down 18 with a four-shot lead it was really cool.
"A lot of these events that I haven't come through on, just means so much more.
"I tried to learn from those situations where I didn't get it done and didn't come through down the stretch, just tried to keep working on my game and keep practicing hard and keep believing that I can do it.
"Took a little longer than I wanted, but I've been playing some good golf for a few years and it's nice to get a W."
A closing 70 saw Henley equal the tournament record with a total of 23 under par, four shots clear of fellow American Brian Harman.
Scheffler, who needed to win or finish solo second to reclaim top spot in the world rankings from Rory McIlroy, finished in a tie for third on 18 under with Ireland's Seamus Power, Joel Dahmen, Troy Merritt and Will Gordon.
Norway's Viktor Hovland, who was bidding to win the tournament for the third straight year, carded a final round of 68 to finish in a tie for 10th.