The 27-year-old ranked 250th in the world was 10-under and in the lead but facing a par putt at his final hole when play was halted for more than an hour because of thunderstorms in the area.
"Luckily, it was a three-footer straight up the hill, so (there) wasn't too much to stress about," Riley said. "I knocked in about five three-footers before walking over there to cap off the round - hit it centre cut and made it.
"It was nice to finish the day and made for a good, pretty stress-free six-under."
Hayden Buckley returned from the delay to card the fifth of his five birdies at the sixth - his 15th hole of the day - posting a five-under par 65,
He was tied for second, two shots back on eight-under 132 alongside Pierceson Coody.
Coody had an eagle and six birdies in his five-under 65, surging to his share of second with three straight birdies to end his round.
South Korea's Im Sung-jae had seven birdies in his six-under 64 to join a group sharing fourth on 134. He was joined by American Keegan Bradley and Austrian Sepp Straka, who both shot 66.
Riley is seeking his second PGA Tour title, having teamed with Nick Hardy to win the 2023 Zurich Classic two-man team event in New Orleans.
One off the lead to start the day, he teed off on 10 and rolled in birdies at 15 and 16 before launching a run of three straight birdies at the 18th.
He got up and down from bunkers for birdies at both the first and second, and rolled in a 10-footer for birdie at the sixth to reach 10-under.
Reigning Open champion Brian Harman headlined a group on 135 and former US Open champion Gary Woodland, back on tour this season after surgery for a brain lesion, was in a group on 136 after a six-under 64.
"I just put everything together," said Woodland, who said he doesn't have the same debilitating symptoms he had last year but is "still battling, still on medication".
Scheffler climbs
World number one Scottie Scheffler was a further stroke back on 137 after a five-under-par 65 that featured five birdies - four in a row from the 18th through the third.
Scheffler was in danger of seeing his cuts-made streak end at 36 after his two-over opening round.
But a week after his arrest in a traffic incident before the second round of the PGA Championship in Louisville, Kentucky, Scheffler turned things around.
Scheffler, who put together a run of four wins in five events - including a second Masters title in April - is still facing multiple charges in Louisville that include felony assault of a police officer after allegedly trying to go around a traffic jam as police investigated an earlier fatal road accident.
Scheffler has called the incident "a huge misunderstanding," with his lawyer saying he "did not do anything wrong but was simply proceeding as directed".
