Henley, the world number four who captured his fifth PGA title at the Arnold Palmer Invitational in March, broke free atop a crowded leaderboard with three birdies to close his round at East Lake in Atlanta, Georgia.
The American had made a strong start with two birdies and an eagle on the front nine, adding back-to-back birdies at 12 and 13 before he drained a 42-foot birdie at 16 to launch his closing burst.
He added a 34-foot birdie at 17 and got up and down from a greenside bunker for birdie at the par-five 18th.
His 207 total feet of putts made included three from beyond 30 feet.
"I putted as good as I can putt," Henley said. "I felt like I was clear on my reads. Last week I felt like I played really well and didn't give myself a bunch of looks because I couldn't figure out how far the ball was going and struggled a little bit on the looks of those greens.
"(Today) just felt a little more clear in my mind on what I thought the ball was going to do ... just kind of free-wheeled it a little bit."
He walked off the 18th green with a three-shot lead, but Scheffler -- who has won five times in his last 10 starts and nabbed major titles at the PGA Championship and British Open this year -- closed with back-to-back birdies in a bogey-free seven-under par 63.
He was a stroke in front of five players, sharing third on six-under 64.
"I hit some fairways and I holed some nice putts as well," said Scheffler, who got his round going with birdies at the second and third and added another at the sixth.
Unhappy with his approach at the 10th, he nevertheless rolled in a 27-foot birdie putt, and after a birdie at 15, managed to get up and down for par from the dangerous right rough below the elevated green at 16.
"I knew going down there right of 16 was a huge penalty," Scheffler said. "We had talked about it in the practice rounds, and our job was just to get the ball back on the green, which I did, and it was nice to hole that long putt."
He drilled another 27-foot birdie at 17, then blasted out of a greenside bunker to four feet at 18 and made that putt for solo second moments before approaching thunderstorms would have halted play.