The 30-year-old even had a birdie putt on the 18th for a 58 which would have equalled the lowest round in professional golf.
Fisher missed that but completed his par to write his name in the European Tour record books.
He bagged 10 birdies and an eagle without a single dropped shot as he soared to the top of the leaderboard, where he was joined by compatriot Eddie Pepperell and Australian Lucas Herbert in a three-way tie for first place ahead of the weekend.
Another Englishman, Matt Wallace, was a shot further behind on 11-under.
Sergio Garcia stayed on track to continue his warm-up for next week’s Ryder Cup with second round 70 that allowed the Spaniard to make the cut for the weekend despite ending his round with three consecutive bogeys.
Since the formation of the European Tour in 1972 there have been 19 rounds of 60.
“It feels great,” said Fisher, whose opening round on the Dom Pedro Victoria Golf Course, was a pedestrian level-par 71.
“I started off with three straight birdies and then kept it going. I’m just pleased that I two-putted there from 40-foot on the last. It was a great day.
“It was fun, I tried to enjoy it and thankfully I got over the line.”
London-based Fisher’s previous claim to fame on the European Tour was winning the Czech Open in 2011 — his only title.
While not unheard of, sub-60 rounds are extremely rare in professional golf.
Furyk, who also has a 59 to his name, carded his 58 in 2016 at the Travelers Championship.