Chaouki Ben Saada was also on the scoresheet to help Tunisia secure their first win after settling for a 2-2 draw with Senegal last Wednesday.
Katlego Mphela scored South Africa's only goal late in the match.
Tunisia are in charge of Group D with four points, leading second-placed Angola on goal difference. Senegal and South Africa lie third and fourth respectively with one point each.
It was South Africa who had the first threat after seven minutes when Sibusiso Zuma's powerful shot from inside the area was blocked by Tunisia keeper Hamdi Kasraoui.
Dos Santos fired Tunisia in front just one minute later when he escaped his marker inside the area and dived to superbly head a right-wing free-kick into the back of the net.
Tunisia fully dominated afterwards and found no difficulty in breaking through South Africa's shaky defense.
Ben Saada extended the lead in the 32nd minute, notching home on the rebound after Dos Santos' fierce low drive was denied by the post.
The Brazilian-born striker heaped more misery on the Bafana Bafana two minutes later.
He pounced on a poor back-pass by defender Nasief Morris before rounding the keeper and slotting into an empty net.
South Africa were outplayed throughout the game, but managed to score a consolation three minutes from time.
Mphela ghosted behind his marker to divert Steven Piennar's mis-hit shot into the back of the net.