Jamieson began the day a stroke behind the former British Open champion, but finished with a 15-under total to lead Oosthuizen (74), Thongchai Jaidee of Thailand (73) and Julien Quesne of France (67).
Padraig Harrington (67) and Francesco Molinari (68) were another two strokes back in the event for European Tour winners.
Jamieson won his first European Tour title a month ago in the season-opening Nelson Mandela Championship. Now he's one round away from earning the "Durban Double.
"It's been another great day. While I didn't play as well as I did the first two days, especially yesterday from tee to green, I managed to score well, which is the main thing," Jamieson said. "So I'm delighted to be in the position I'm in.
"The experience of winning the Nelson Mandela in a playoff under the extreme pressure with a first European Tour title on the line gave me a lot of confidence that I've taken into this week."
Jamieson had an eagle on the par-5 eighth hole after landing the ball 5 feet from the hole with a 5-iron for 198 yards. He also birdied three straight holes from the 11th.
For three days, Jamieson has competed against some of the bigger names in golf.
He played the first round with Harrington and finished a shot ahead
"You always have your eye out, comparing yourself to them," Jamieson said. "But yeah, I managed to beat them through the week, so that was nice."