Henley sank 15-foot birdie putts on the par-three 16th and par-four 17th holes, then birdied the 18th from just inside 17 feet to force a playoff with Cole at Colonial Country Club in Fort Worth, Texas.
On the first playoff hole, the 18th, Henley dropped his approach just inside five feet and, after Cole missed a birdie putt from just inside 13 feet, sank the tension-packed winning putt.
"Felt like I was hitting good putts most of the day and they just went in at the end," Henley said.
"Then to come back to the playoff and do that, I'm still kind of shaking. That was as nervous as I've been over a putt in my whole life."
Henley, who shared third at the Masters in April, captured his sixth career PGA Tour victory and first since last year's Arnold Palmer Invitational.
"I just kept telling myself, 'I want to win,'" Henley said. "I want to be hitting these putts and be in contention. This is why I practice hard.
"I feel like I've worked harder and harder and just been a little off mentally this year. Just fought really hard until the end. It felt good to see an awesome result."
Henley fired a three-under par 67, and Cole shot 70 to finish 72 holes deadlocked on 12-under 268.
Cole overcame a double bogey at the ninth hole with a 24-foot birdie putt at the par-five 11th to regain the solo lead until Henley's closing run, missing out on a prime chance for his first PGA Tour triumph.
"It's very disappointing," Cole said. "I felt like I played pretty well for the most part. Hung pretty tough when I did miss a few shots. It's disappointing, but I played well, and I'm proud of that."
Defending champion Ben Griffin and fellow Americans Alex Smalley and Mac Meissner shared third on 269 with a sixth-place pack on 270, including reigning US Open champion J.J. Spaun, fellow Americans Gary Woodland and Michael Brennan, and Colombia's Nico Echavarria.
Henley opened eagle-birdie, made bogeys on the next three holes, and was only lurking until his late flurry.
"I was feeling a little jittery or quick or something on the front," Henley said. "I just kind of calmed down a little bit and started to hit some good shots."
Cole, the son of South African pro golfer Bobby Cole and retired US LPGA star Laura Baugh, led by one stroke when the day began and made a birdie-birdie start.
Cole found water on his approach at the ninth hole and made a double bogey, but when Spaun and Brennan made early back-nine bogeys, and Cole sank his clutch birdie at 11, he was back at the top until the last hole.