"It's always nice to have a good start," Fisher said after his round, which included six birdies on his first nine holes.
Jeppesen had two eagles and two birdies on his front nine.
"It was a good round and I'm very pleased," the 24-year-old Swede said.
Michael Hoy held third place at 7-under 64.
Defending champion Richard Sterne was far back on the leaderboard after shooting an even-par 71.
The event is co-sanctioned by the European Tour and the Sunshine Tour of South Africa.