Furyk hit a wedge shot to 3 feet on the par-4 ninth, his 18th hole, to become the sixth player in PGA Tour history to break 60 in an official event.
'It's a magic number,' Furyk said. When asked if it was his greatest-ever round, he replied: 'Absolutely.'
After opening in 1-over 72, Furyk now shares the 36-hole lead with Brandt Snedeker (68) at 11 under.
Beginning on the back nine at Conway Farms, Furyk birdied four of his first five holes and then holed his approach shot for eagle on the par-4 15th. He wrapped up his opening nine with back-to-back birdies for an 8-under 28.
After a par on the first hole, his 10th, Furyk birdied Nos. 2, 3 and 4. At that point, Furyk was 11 under through 13 holes and needed to play his final five holes in 2 under to become the first-ever player to shoot 58 in a Tour event.
Furyk, who was recently snubbed by U.S. captain Fred Couples as a Presidents Cup wild card, three-putted for bogey at the par-4 fifth but recovered with birdies on Nos. 7 and 9. His previous best score on Tour was 62 (five times).
'Just said to myself on (No.) 9, how many opportunities are you going to have in your life to shoot 59,' Furyk said.
'I'll enjoy this for a few hours at least. ... There is still a tournament to try and win.'
The 16-time PGA Tour winner is seeking his first Tour title since the 2010 Tour Championship, which earned him FedEx Cup champion honors.