The 43-year-old American became just the sixth man to break 60 on the PGA Tour, which has seen him tie the lead at the BMW Championship on 11-under-par, alongside 2012 Ryder Cup teammate Brandt Snedeker.
"It was a little hectic down the stretch," Furyk told NBC. "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. I told myself at nine, 'how many chances are you going to have in your life to do this again?'.
"It's absolutely the best round of golf I've ever played."
The last man to post 59 was Australia's Stuart Appleby at the Greenbrier Classic three years ago.