Fleetwood fired four birdies and a bogie in his round of 69, giving him the slender advantage over France's Victor Dubuisson and Joost Luiten of the Netherlands.
A triple bogey on the 16th cost home favourite Louis Oosthuizen the overnight lead but he will start Sunday's final round two shots behind Fleetwood and still in the hunt.
Luiten, who shot the first albatross of the European Tour season on Friday at the par-five 10th hole, Oosthuizen and Fleetwood were joint leaders going into Saturday's round
The 22-year-old Fleetwood, who won his only other European Tour event when he claimed the Johnnie Walker Championship in a playoff at Gleneagles last August, went out in 33 on the front nine, but had to wait until the 18th for his next birdie having dropped a shot on the 14th.
"I had to rely on short game today. I scrambled it round well and that's satisfying," the Englishman told reporters.
Dubuisson carded a 69 after a strong back nine in which he managed three birdies, while Luiten shot 70 but fluffed a six-foot putt for an eagle at the final hole.
Former Open champion Oosthuizen's fortunes swung wildly, with seven birdies, three bogies and the triple bogie on the last for a one-under-par 71.
The 2011 winner, Englishman Paul Casey, carded a 65, the low round of the week, but remained six shots behind the leaders on four under par for the tournament, the same score as Irishman Padraig Harrington and England's Darren Clarke.
The elite 36-man field is made up of winners on the Tour over the last 12 months plus one former tournament champion and five multiple tour event winners given special exemption.