The South African began the final round at Royal Johannesburg and Kensington Golf Club four shots off the pace, but produced a fine round to reach 19 under and claim victory.
Tyrrell Hatton, Jin Jeong and Justin Walters shared second place, the former matching Coetzee's six-under 66 on Sunday.
Walters could only muster a closing 73, while the other overnight leader, Thomas Aiken, went round in 74 to finish tied for fifth on 15 under.
Coetzee climbed into contention with four front-nine birdies and picked up further shots on the 10th and 15th in a bogey-free round.
The 27-year-old, who becomes the sixth South African to win the event, birdied all three par-fives, and did well to save pars at the 11th and 12th after running into trouble.
"I don't know what I'm feeling," said Coetzee, whose victory came at the 107th time of asking and after 24 top-10 finishes.
"I'm lost for words. I’ve been waiting a while and I started doubting, so I'm very happy.
"I got to share this with my family and my friends. It was my mum's birthday yesterday (Saturday) and I wanted to do it for her as I forgot to buy her a present!"
Coetzee's win guarantees him a place at the British Open later in 2014.
Walters and Jeong have also earned spots in the tournament at Royal Liverpool, with Hatton missing out due to his inferior world ranking.