The 26-year-old edged ahead of compatriot and home favourite Charl Schwartzel who is seeking his first victory in the event.
A steady front nine saw Basson record two birdies to turn in 34 but he picked up four shots in five holes, culminating with that eagle on the 500-yard 15th to finish six under.
Starting at the 10th Schwartzel, the 2011 Masters champion, was just two under through 14 holes but a brilliant iron tee shot to five feet at the 182-yard sixth sparked a run of three birdies in four holes.
His drive at the next required a drop but a chip to a foot allowed him to save par and birdie putts of 12 and 15 feet saw him finish bogey-free on five under to move into second place.
"It means a lot, I'd love to win the South African Open," Schwartzel told www.europeantour.com.
"I did my preparation and figured out the way I want to play the golf course and it worked today.
"I've always had really good results in the South African Open, come close a few times.
"I reckon it's one of those where you have to keep at it and keep playing and it will come eventually."
England's Simon Dyson, playing his first tournament since disqualification from the BMW Masters after signing for an incorrect score, shot a four-under 68 to move into a share of third with compatriot Steve Surry.
Dyson put aside concerns over an impending disciplinary hearing after being charged with a serious rules breach, for which the most severe punishment could be a ban, despite insisting his actions were an accidental mistake after television viewers spotted him touching the line of a putt.
However, his round was erratic with his opening nine holes including five birdies and two bogeys and, although he started the back nine with another dropped shot, he recovered with two more birdies to post 68 - the same score as another South African, Warren Abery.