Anderson, 31, had to come from behind though to bag the win over the clay-court specialist.
After dropping the first set 6-2, the big serving South African had to dig deep for some good old Mzansi fighting spirit and won the second set 7-6.
Anderson’s fightback was briefly halted when rain interrupted play for a while.
But once the two players took the court again, it was all about Anderson as he won the third set 7-6, before cruising to a fourth set win (6-3).
Top-ranked home favourite Andy Murray made light work of his first round opponent, Alexander Bublik, 20, beating the rookie in straight sets 6-1, 6-4 and 6-2.
Another crowd favourite, Rafael Nadal, meanwhile, also cruised to the second round when he beat Australian John Millman in straight sets 6-1, 6-3 and 6-2.