Playing without a practice round, Fisher relied mainly on the advice of caddie Adam Marrow, who had walked the course twice.
"He told me where to hit it, where the run-offs were and what clubs to hit. He was pretty much spot-on," Fisher said.
The 27-year-old Englishman started at the 10th hole on the 7,257-yard London Golf Club course and birdied four of the first nine, with his only bogey at the eighth.
He went on to birdie his last six holes, capping his run with a wind-assisted drive of more than 400 yards at the par-4 ninth, finishing about 35 yards short of the flag.
Rory McIlroy was 5-under, and a group of seven was another stroke back.
Fisher broke the course record of 64 set by Seve Ballesteros when the club was opened in 1994.
Fisher described his round as "a bit surreal" after deciding against a practice round Tuesday and settling for light practice Wednesday because he was exhausted after his successful 36-hole qualifying bid Monday for the British Open.