The 24-year-old feared for his overnight five-shot lead going into the third round at Carnoustie, the toughest of the three courses being used in this week's $5 million tournament.
Grace, however, fired a three-under-par 69 to finish on 20-under 196, four strokes clear of Dane Thorbjorn Olesen (68) ahead of Sunday's final round at St Andrews.
Swedish pair Alexander Noren (65) and Fredrik Andersson Hed (66) were a further shot adrift on 201.
Grace made six birdies but was not surprised to run up a double-bogey six at Carnoustie's fearsome 15th hole.
"I didn't play particularly well but I got it round," he told reporters. "But this course does that to you mentally.
"You expect to hit a couple of bad shots out there but I kept fighting. Everybody calls it 'Carnasty' and that name for me says it all - for me it's a lot tougher than the rest."
Having beaten fellow South Africans and multi major winners Ernie Els and Retief Goosen in a playoff to land the Volvo Champions title in his homeland in January, Grace is not worried about leading from the front in the last round.
"I think if I keep hitting the ball the way I am and make the putts, anything is possible," he said.
"But Thorbjorn is one hell of a player and he has shown in the past that he can play with the big names as well. He is not going to fall over under pressure."
Sicilian Open champion Olesen, playing his third round in the Dunhill Links at Kingsbarns, was within two shots of Grace at one stage but slipped back after a double bogey at the 17th.
British Open winner Els, who gave Grace a place in his golf foundation when he was a teenager, showered praise on the leader who has also won this year's Joburg Open and China Open.
"Branden has always been a quality player but he has really come on now," said Els.
"Fortunately for him he hits it nice and low and it runs for ever. He could well even win an Open championship, he's that good."