The off-spinner claimed seven second innings wickets while leg-spinner Amit Mishra grabbed the remaining three as South Africa, chasing 310 to stay alive in the contest, folded for 185.
The setback also meant South Africa, currently the top ranked team in this format, succumbed to their first away test series defeat since 2006.
Ashwin, who had claimed five wickets in the first innings, made the most of the turning track at the VCA Stadium, dismissing Dean Elgar (18) and AB de Villiers (nine) in the morning to peg back South Africa.
South Africa skipper Hashim Amla dug in to negate the spin, playing with soft hands so edges did not carry to slip, while Faf du Plessis used his feet well against the spinners.
In a test match that saw 32 wickets tumble on the first two days, the pair showed grit and determination to thwart the Indian bowlers for more than 46 overs with the biggest partnership in the low-scoring contest.
The duo survived numerous leg-before and bat-pad catch appeals before Mishra ended the 72-run stand that unhinged the visitors.
Amla offered an edge off the top half of his bat to perish in gully and in Mishra’s next over, Du Plessis lost his stump to a ball that kept low. Both batsmen contributed 39 runs each.
Ashwin then returned to mow down the Proteas tail to finish with a match haul of 12 wickets that also earned him the man-of-the-match award.