Furyk capped his round at the par-71 Conway Farms Golf Club with a three-foot birdie at the ninth hole, his last of the day.
He had 11 birdies, one eagle and one bogey and joined Brandt Snedeker atop the leaderboard yesterday.
"Pretty exciting day," said Furyk, 43.
American Furyk got to six-under through his first six holes with four birdies and an eagle at the par-four 15th, where he holed a 115-yard wedge.
He birdied 17 and 18 to complete a back-nine 28. Furyk, a former US Open champion and eight-time Ryder Cup player, said he tried to forget about his first nine holes as he set off on his inward run, knowing he needed four birdies to break 60.
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"It was a little hectic down the stretch," Furyk said. "I had the three-putt at five and you could see the nerves at the sixth, leaving the putt short.
"I got myself back on track at seven, then I was kind of between clubs at eight with a pretty good chance to make my birdie."
After he didn't get that birdie at the penultimate hole, "I told myself at nine, 'how many chances are you going to have in your life to do this again?'"
Furyk is the first US PGA Tour player to post a 59 since 2010, when both Paul Goydos and Australian Stuart Appleby achieved the feat.
David Duval, did it in 1999, Chip Beck in 1991 and Al Geiberger was the first to card a 59 on the US tour in 1977.