Defending champion Loren Roberts and two-time Masters champion Ben Crenshaw share second place at minus-7. Hale Irwin, the 1997 winner, opened with a 6-under-par 66. He was joined in fourth by Allen Doyle and Jerry Pate.
Bryant got off to a flying start with four straight birdies. He moved alongside Crenshaw at minus-5 with a birdie at the sixth. Bryant added another birdie on seven to climb into the lead at minus-6.
The 52-year-old converted a birdie at the 10th to again move one stroke clear of Crenshaw. Bryant got to minus-8 thanks to an 18-foot birdie putt on the 13th.
He two-putted for par on each of the next four holes. Bryant closed in remarkable fashion as he drained a 58-foot birdie putt at the last to tie his career-best round. He previously carded a 63 en route to winning the 1995 Walt Disney World/Oldsmobile Classic on the PGA TOUR.
'It felt amazing. I kept hitting these putts that kept going in. It was the single best putting round of my life,' said Bryant. 'I've never had an experience quite like that. Even at the last hole, it was just magic.'
And what about that long birdie putt at the last?
'Well Tony made a good read,' Bryant said of his caddie. 'We had 58 feet and it was dead straight. Both of us just said straight and it went down there and the ball went straight. I don't know what happened there, it just disappeared.'
Roberts, who fired a course-record 61 in the final round last year to win, also got off to a quick start. He birdied three of the first four holes. A birdie at the seventh helped Roberts turn at 4 under.
He picked up his fifth birdie on the par-5 10th. Roberts got up and down for birdie on the par-5 14th. At the last, he poured in a 20-footer for birdie to share second.
'Believe me, if you can get the ball started on line here, they go in,' Roberts stated. 'There's not a whole lot of movement. Obviously the scores reflect that. The golf course really just sets up well. For me at least, I can see all the tee shots and exactly where I want to hit them. I can just visualize the shot so well.'
Crenshaw didn't collect his first birdie until the fourth. His other front-nine birdie came at seven. Around the turn is where the two-time Masters winner did his damage.
The 55-year-old Crenshaw, who has not had a top-three finish on the Champions Tour, ran off three straight birdies from the 10th to jump to minus-5. He drained a 7-footer for birdie on 14 to tie Bryant in the lead.
Crenshaw dropped his approach shot within 8 feet at the 18th. He sank that birdie putt for his share of second.
'The conditions are wonderful and the course is great. I'm here on a sponsors exemption...so I am very lucky that I am here,' Crenshaw admitted. 'I played well and hit the ball pretty solid.'
Dana Quigley, who won here in 2003 and 2005, posted a 5-under-par 67. He shares seventh place with Tom Jenkins, 2002 winner Tom Kite, 1998 champ Gil Morgan, Eduardo Romero, Craig Stadler and D.A. Weibring. Six more players, including Tom Watson, are one stroke further back at minus-4.