Stroud broke out of a three-way tie atop the leaderboard, finishing 54 holes at Coco Beach Golf and Country Club with a 15-under par total of 201.
He was one shot in front of fellow Americans D.A. Points and Bill Lunde, who had shared the second-round lead with Bryson DeChambeau and both carded 69s.
DeChambeau fired a 70 to join a group of six players sharing fourth on 203 that also included England's Andrew Johnston, South Korean Kim Whee, Puerto Rican Rafael Campos, New Zealander Tim Wilkinson and American Jonathan Randolph.
Stroud, who started the third round one shot off the lead, had five birdies without a bogey to put himself in position to challenge for a first US PGA Tour title Sunday.
His previous best finish was a playoff loss at the Travelers Championship in 2013.
"I feel like I'm way overdue," Stroud said. "I'm playing very, very well. I'm making some nice putts."
He said his approach on Sunday would be simple.
"Nothing fancy -- just go out there and hit a lot of fairways, a lot of greens. There's a lot of guys behind me. I'm just going to keep looking forward."
Points had four birdies and a bogey in his round, but couldn't find a birdie after the 12th.
Lunde, seeking a second US PGA victory almost seven years after capturing his first, had joined the second round leaders with three pars to end his weather-disrupted 68 on Saturday morning.
He followed with a three-under effort that included five birdies and two bogeys.
Johnston stormed up the leaderboard with a bogey-free six-under par 66. His six birdies included two to cap his round.
Whee carded a 67 and Randolph a 68. Wilkinson followed up a second-round 63 with a 69 and Campos, who had seized sole possession of the lead with five birdies on the front nine, settled for a 69.
The tournament offers a chance for lesser lights on the PGA Tour to contend while top-ranked talent plays at Austin, Texas, in the World Golf Championships Match-Play tournament.