ST ANDREWS, United Kingdom, Oct 7, 2012 (AFP) - Grace, who won for the fourth time on the European Tour this year, closed with a two-under 70 over the Old Course, with the fast-rising Olesen coming in with a 68.
The 24-year-old from Pretoria began the week with a European Tour record-equalling 60 at the nearby Kingsbarns course - the European Tour's pro-celebrity event is staged over three links courses - and was never overtaken after that.
Olesen did make a move in the final round, drawing level with seven holes to go, but the South African birdied the next three holes and with another on the last finished with a tournament record-equalling 22-under-par total.
Alone in third place, four strokes off the pace, was Alexander Noren of Sweden, while another Swede, Joel Sjoholm, was fourth a further two strokes back.
Scotland's Stephen Gallacher, winner in 2004, finished in a tie for fifth after incurring a two-stroke penalty for playing the wrong ball at the 16th.
Grace, who became the first wire-to-wire winner on The European Tour season, is a product of the Ernie Els Foundation, following in the footsteps of Louis Oosthuizen, who won the British Open at St Andrews in 2010, and 2011 Masters winner Charl Schwartzel.
Winning at St Andrews, as Oosthuizen did in the Open, was especially emotional for Grace.
"I've really dreamt of this moment my whole life," he said. "I had goosebumps thinking this morning about Louis and the possibility of holding a trophy here myself," he said.
"It was a tough day, but the putter started working and that's all I needed to do."
There were celebrations too for another South African as George Coetzee carded a 10-under-par 62 to tie for the St Andrews Old Course record.
The 26-year-old, having only just made the cut with a 15 foot birdie putt on his last hole at Kingsbarns on Saturday, fired 10 birdies in a flawless final round - including a stretch of eight in nine holes - to surge 58 places up the leaderboard into a tie for fifth.
It would have been a course record in its own right were it not for the first day heroics of Frenchman Victor Dubuisson, who deposed Luke Donald, Simon Dyson and Rory McIlroy as the former incumbents of the record with a 62 of his own on Thursday.
"Every kid dreams about it, it's the home of golf," said an elated Coetzee.
"Every time I step up, it's just an unbelievable feeling. I'll play the course any way they let me, and obviously 62, I'm very happy, especially to move up into the top ten only just making the cut yesterday.
"I had five birdies on the trot and I missed one. I think I lipped out one on 16 and if that one had gone in it would have been nine on the trot.
"But it's one of those days, you can always count the shots that you've left out there but all in all, I'm very happy with a 62."
Collated final round scores on Sunday in the European PGA Tour Alfred Dunhill Links Championship at the Old Course St. Andrews, Scotland:
(GBR and Ireland unless stated, par 72):
266 - Branden Grace (RSA) 60-67-69-70
268 - Thorbjorn Olesen (DEN) 63-69-68-68
270 - Alexander Noren (SWE) 64-72-65-69
272 - Joel Sjoholm (SWE) 65-67-70-70
273 - Stephen Gallacher 67-70-65-71, Danny Willett 66-69-68-70, George Coetzee (RSA) 71-72-68-62, Victor Dubuisson (FRA) 62-73-70-68, Fredrik Andersson Hed (SWE) 64-71-66-72, Raphael Jacquelin (FRA) 69-67-68-69, Lee Slattery 67-70-68-68
274 - Steve Webster 70-72-68-64
275 - Ricardo Gonzalez (ARG) 71-72-63-69, Magnus A Carlsson (SWE) 66-68-73-68
276 - Anton Haig (RSA) 65-68-73-70, Chris Wood 69-70-68-69, David Drysdale 66-72-70-68, Mark Foster 72-69-66-69, Thongchai Jaidee (THA) 65-71-67-73, Graeme Storm 68-66-72-70, Oliver Fisher 66-74-67-69
277 - Brett Rumford (AUS) 71-70-69-67, Thomas Aiken (RSA) 72-68-67-70, Mikko Ilonen (FIN) 72-71-66-68, Ross Fisher 70-69-69-69, James Morrison 70-69-71-67, Ashley Hall (AUSs) 64-73-69-71, Edoardo Molinari (ITA) 71-68-70-68
278 - Emiliano Grillo (ARG) 70-69-67-72, Paul Waring 69-66-70-73, Gonzalo Fdez-Castano (ESP) 70-69-70-69, Darren Fichardt (RSA) 71-66-72-69, Padraig Harrington 71-73-66-68
279 - David Howell 71-72-67-69, Oliver Wilson 67-71-69-72, Martin Kaymer (GER) 70-71-69-69, Mathias Gronberg (SWE) 74-66-69-70, Maarten Lafeber (NED) 71-71-65-72, Scott Jamieson 68-74-68-69, Peter Hanson (SWE) 72-70-69-68, Marc Warren 70-70-69-70, Shane Lowry 69-69-71-70, Ernie Els (RSA) 73-70-65-71, Alejandro Canizares (ESP) 66-70-71-72, Lorenzo Gagli (ITA) 69-67-72-71, Bradley Dredge 69-69-72-69