He's two shots ahead of Sweden's Henrik Stenson and South Africa's Matthew Carvell at Serengeti Golf and Wildlife Estate on the outskirts of Johannesburg.
Former Masters champion Charl Schwartzel recovered from a double-bogey 5 on his third hole at the 12th for a 68. Martin Kaymer shot 70, while Branden Grace was even par.
The 26-year-old Bremner consolidated his outstanding 30 with two more birdies on the back nine. His bogey-free round raised hopes of a first European Tour title at his home event.
"I played fantastically on the first nine," he said. "I didn't miss a green and there was only one chance that I missed that was makeable, and that was on 15. But in theory, I could have shot 28 on the first nine."
Craig Lee and Alex Haindl each shot a 67 to trail by three strokes.
After his struggles on No. 12, Schwartzel returned to the form that landed him the 2011 title at Augusta National. He had a chip on No. 2 to set up one of his six birdies and didn't drop any more shots.
Kaymer double-bogeyed his first hole on his South African Open debut—calling it a "bit of a shocker"—but picked himself up with four birdies.
Grace, enjoying a breakthrough season since winning back his tour card for 2012, could only manage three birdies, plus a bogey and double on his back nine. Defending champion Hennie Otto shot 71 and amateur Terence Boardman made a hole-in-one on the par-3 12th and finished with a 74.