Oosthuizen carded a 4-under-par 69 while Brier, who bogeyed the 16th, hit 70 as they finished the third round at 13-under 206, a stroke ahead of defending champion Charl Schwartzel and three others on a packed leaderboard.
Twelve players were within three strokes of the leaders at 6,104-meter (6,674-yard) East London Golf Club.
“This course needs the wind,” said Oosthuizen after a day without strong winds. “I hope the wind comes up a bit tomorrow, and then we can get back to the golf we played on the first two days.”
Branden Grace of South Africa, who shared the overnight lead with Brier and Miles Tunnicliff of England, appeared to be sailing serenely to a comfortable lead going into the final round before he misjudged his second into the 16th, going through the green. Twice in succession, he was unable to get a club on the ball as he swung at it buried in thick grass.
The resulting triple-bogey 7 saw him slip out of sole possession of the lead at 15 under to 12 under with two holes to play.
Ahead of him, however, Oosthuizen, who had inherited the sole lead at 14 under, had to take a penalty drop when he hit his tee shot into unplayable rough on 18. He was unable to salvage par, slipping to 13 under for the tournament co-sanctioned by the European and Sunshine Tours.
Grace was clearly unnerved by the problems he had on 16, and pulled his tee shot on the 152-meter (166-yard) 17th right.
His ball came to rest against a tree, and his second ended up on a paved path. He got his third out to within 12 feet of the pin, but he finished with a double-bogey 5 and found himself at par for the round at 10-under, where he started the day.
“I played great golf for 16 holes,” Grace said. “The last three holes just bit me at the end. It was a couple of funny decisions I made and a couple of wrong clubs, but all in all, I’m happy to be in contention.”
Brier, chasing his first European Tour title since the 2007 China Open, dropped his only shot of the day on 16 when he, like Grace, overshot the green. But he was able to recover enough to drop just a single shot, and, with two birdies and an eagle on his card, the damage was not too bad.
Tunnicliff shot 72 to fall two strokes off the pace.
Schwartzel appeared to be struggling for the early part of his homeward nine as he dropped shots on the 10th and the 12th, but he birdied four of his last five holes to card a 68 and move to 12 under overall.
Jaco van Zyl, a four-time winner on last year’s Sunshine Tour, didn’t make a single birdie on his homeward nine but joined countrymen Schwartzel, Jbe Kruger (67) and Spain’s Manuel Quiros (68) a strike off the lead.
The round of the day came from England’s Robert Dinwiddie, who, like Van Zyl, was bogey-free during his 7-under 66 which lifted him to 10 under.