Grace, who has won three European Tour events this year, took full advantage of the superb scoring conditions at Kingsbarns, one of three courses hosting the $5 million event.
The 24-year old Grace's score was two shots better than the 62 posted by England's Lee Westwood on his way to victory in 2003.
While no player has recorded a 59 in the 40-year history of the European Tour, Grace is the 15th player to shoot 60.
"It could have been a 59, and I said to my caddy when I hit it stiff on 18 that it could be close to a 59," Grace said. "I had some opportunities, but you know, I never really made anything long. But that round has to be my best ever."
Victor Dubuisson shot a course record 10-under 62 on Old Course at nearby St. Andrews, despite dropping a shot at the 16th hole.
"It's very special as I had 10-under par in Italy last year, but today I could have done better," Dubuisson said. "I'm not saying that 10 under is really bad. It's my best score I have ever played, but it's just that my longest putt was (16 feet)."
Dubuisson's round is one stroke less than the previous low of 63 set by three players, including Northern Ireland's Rory McIlroy during the 2010 Open Championship.
The lowest round of the day at Carnoustie was 67, shared by England's Oliver Wilson and Frenchman Gregory Havret.
The four Ryder Cup players in the event struggled at Carnoustie. Martin Kaymer shot 70, American Dustin Johnson and Peter Hanson had 72s. Johnson moved to 4 under after 12 holes but dropped four shots in his closing four.
Paul Lawrie finished with a 75.
"I am a bit tired and just played poorly," said Lawrie, winner of the inaugural event in 2001. "My first 12 holes was probably my worst ball striking of the year but I put that down to a bit of jet lag, and I also didn't sleep well last night."