After sinking a 10-foot birdie putt at his penultimate hole, the par-four eighth, he needed another birdie at the ninth to fire a 59 at the Sedgefield Country Club in Greensboro.
After an aggressive drive ended in a "tough lie" in the left rough, Kim's approach from 112 yards ended 50 feet beyond the pin and his hopes of holing the birdie putt appeared slim.
But he judged the break and the speed of the downhill effort almost perfectly, his ball grazing the hole before settling within tap-in distance.
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Kim's round in ideal morning conditions lifted him into the second-round lead at 12-under 128, while Furyk (64) continued his sizzling form to jump into a tie for second at 10-under with fellow Americans Kevin Na (67) and Lucas Glover (61), and Japanese Hideki Matsuyama (64).
Glover also had a chance to shoot 59, but the former U.S. Open champion three-putted the last after leaving his 27-foot birdie putt 18 inches short.
Joint first-round leader Rafa Cabrera-Bello, of Spain, shot 68 to slip three shots behind.
Kim said he did not really think of a 59 until his birdie at the eighth had followed an eagle three at the fifth.
He added that his putting and iron play were particularly sharp.
"Yesterday the greens were firm and fast but today ... much better, soft," he said.
Kim, 21, is 43rd on the FedExCup points list, thanks largely to his playoff defeat by Aaron Baddeley at the Barbasol Classic in July.
While Kim had no regrets about the way he played the final hole, Glover could only rue leaving short his putt for 59.
"I had the read," he said. "I knew it was into the grain up that hill. A lot of long ones I was leaving short but, in the back of my head, I was saying, 'Get it there, get it there, get it there.'
"(I am a) little mad at myself for leaving that one short."
Glover blamed a subsequent loss of concentration for his missed short par putt.
There have been seven sub-60 rounds on the PGA Tour, headed by Furyk's 58. The American has also returned a 59 along with Al Geiberger, Chip Beck, David Duval, Paul Goydos and Stuart Appleby.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Tony Jimenez and Mark Lamport-Stokes)
