Michael Hoy was a shot back at 7-under 64, while Rafael Cabrera Bello, David Drysdale, Alfredo Garcia-Heredia, Andrew McLardy, Eduardo Molinari and Patrik Sjoland were all tied at 6-under 65 on a tight leaderboard.
There were 48 players overall within five shots of the co-leaders.
Fisher, a 29-year-old South African, set an early pace when he teed off in a morning group on the West course ' the East course at Royal Johannesburg/Kensington is also being used ' and made six birdies over his first nine holes.
'It's always nice to have a good start,' he said.
Fisher's only bogey came at No. 10, slowing the three-time Sunshine Tour winner's pace. But he went on to make a birdie at the 11th and back-to-back birdies at the 14th and 15th to secure the early clubhouse lead.
'I had nice rhythm and I putted really well,' said Fisher. 'I gave myself some nice opportunities. I was quite patient, which is a good thing for me as I don't feel rushed, and maybe I can maintain it now for three more days.'
Jeppesen, also playing the West course, teed off in the afternoon and nearly matched Fisher's flying start with a 5-under 31 on the front nine. It included two birdies and eagles at the second and seventh holes.
He knew the course was yielding good early scores.
'Before I went out there I saw that there were a lot of guys shooting good scores on the first six holes,' said Jeppesen, a 24-year-old from Sweden.
He bogeyed the 10th, but completed a mistake-free back nine that included birdies at the 11th, 12th and 15th holes to tie Fisher for the first-round lead.
'It was a good round and I'm very pleased,' said Jeppesen.
Further down the leaderboard, defending champion Richard Sterne was tied for 109th place following an even-par 71. Sterne, who claimed a pair of titles before the holiday break, is attempting to win three consecutive events.
The tournament, co-sanctioned by the European Tour and South Africa's Sunshine Tour, features a field of 210 players.