France's Victor Dubuisson was third after going around the Gary Player Country Club in 70 shots, for a nine-under par score.
Dubuisson moved alongside Jamieson with a four-foot birdie at the first, only for the Scot to edge back ahead by matching Grace's gain at the par-five next.
Last year, Sweden's Alex Noren came from a similar shot deficit to win "Africa's Major" and Fleetwood will need to do the same.
South Africa's Haydn Porteous posted a five-under 67 to be at three-under, while Race to Dubai leader Tommy Fleetwood hit back after a hard first two days to also post a 67.
Champion Branden Grace is sprayed with champagne by Louis Oosthuizen and Darren Fichardt on the 18th green.
"I knew I had to stay patient and take the chances when they came my way, but the key one was 16".
With a final round of 66, Grace won "Africa's Major" by a single stroke on 11 under par and ended a win drought of South Africans in this event stretching back to Trevor Immelman's triumph in 2007. I try to play to the middle of the green.
"I missed some good opportunities, but the problem is I felt like all day I had tricky, downhill putts, and I couldn't really go for it because I didn't want to".
"Today was also not great on the front, so I was happy to be under par (one-under 35) at the turn. I'm very excited to be in this position. hopefully it's third time lucky for me". "But I kept hitting the ball great and giving myself chances".
"I've played with some of the best in the world this week and proved I can compete", he said.
Dual Ryder Cup star Francesco Molinari produced the lowest round of the day with the Italian managing seven birdies in his bogey-free 65 to share fifth place on three-under par.
Jamieson will be joined in that event by 29th-ranked Richie Ramsay and David Drysdale (54th), with Marc Warren agonisingly missing out by one spot after he slipped to 63rd following a disappointing finish in South Africa.
He was three under at the seventh when he secured another birdie but a double-bogey on the very next hole cancelled out all of his good early work.
He shot a final-round 68, which included five birdies and a bogey to be at five-under overall and a tied-sixth finish.