Bryant's score broke the single-day course record on the La Cantera Golf Club course and left him at 16-under 194 after three rounds.
The round tied the 60 shot by Robert Gamez in January at the Bob Hope Chrysler Classic.
The previous mark at the Texas Open was 61, set by Garrett Willis in 2002 and matched by Ted Purdy in Thursday's opening round.
Mahan (62) was at 197, while Joe Ogilvie shot a 61 to join Pat Perez (63), Todd Fischer (63), Tim Clark (64), Patrick Sheehan (65), Scott Simpson (66), Ryan Palmer (67) and J.J. Henry (67) at 198.
Henry was tied with Mahan, but he lipped out a 10-foot par putt on the 18th for his second bogey of the day.
Second-round leader Dean Wilson and Heath Slocum (64) were another shot back at 199. Wilson shot an even-par 70 Saturday.
Bryant, who started the day well back at 6 under, birdied six of his first eight holes for a 29 on the front. On the 7th and 8th, both par-4s, he hit 90-yard approach shots 5 feet from the hole for birdies. On No. 9, he left a 10-foot birdie putt a few inches short.
After birdies on three of the first five holes on the back to get to 15 under, Bryant ran off four straight pars. It looked like another par on the par-4 18th, but he made a 29-foot putt for his 10th birdie of the round.
Mahan, who began the day at 5 under, birdied the par-5 1st, and then he made par on six straight holes before sinking a 19-foot putt for birdie on No. 8 and a 10-footer for birdie on the 9th.
He heated up on the back nine, making birdies on four consecutive holes beginning with a 16-foot putt on the 11th. He fared well on both of the par-3s on the back side by hitting within 5 feet on the 13th and 3 feet on the 17th.
Ogilvie started Saturday at 3 under, just one stroke above the cut. He mixed pars and birdies through his first 14 holes before finishing with three birdies on his last four holes for a 61.