The Proteas, who resumed at 32-2 chasing a target of 310, folded up for 185 after tea on the third day on a dusty, deteriorating pitch at the VCA stadium in Nagpur.
Off-spinner Ravichandran Ashwin bagged 7-66, taking his match haul to 12 wickets after claiming 5-32 in the first innings.
India have now won eight of their last nine Tests on home soil -- all inside three days -- having routed Australia 4-0 and the West Indies 2-0 in 2013 before taking the first and third Tests in the current series.
The winning sequence was broken by the rain-hit second Test in Bangalore where the last four days were washed out.
The defeat was top-ranked South Africa's first series loss abroad in nine years after the 2-0 setback in Sri Lanka in 2006.
India won the first Test in Mohali by 108 runs before the rain-hit second match in Bangalore was drawn.
The fourth and final Test will be played at New Delhi's Ferozeshah Kotla stadium from December 3.
Brief scores:
India 215 and 173; South Africa 79 and 185