India took a winning 2-0 lead in the four-game series by bowling out South Africa for 185 in the last session of the third day.
Ashwin's offspin took 7-66 in the second innings, and 12-98 in the match, to once again lead the charge against a batting lineup that folded for a meagre 79 in the first innings.
South Africa lost a series away from home for the first time in nine years.
Tormented on spin-friendly tracks, South Africa lost this test much like it did in the first one in Mohali, where it was beaten by 108 runs, also in under three days. The second test at Bangalore was a washed-out draw.
The fourth 10-for in 31 tests for Ashwin was supported by legspinner Amit Mishra, who accounted for South Africa high-scorers captain Hashim Amla and Faf du Plessis, who made 39 runs apiece.
Set 310 to win, their defiant 72-run, fifth-wicket partnership lasted 46.2 overs until Amla was caught by captain Virat Kohli at gully off Mishra. Mishra then bowled du Plessis, who went for a pull shot but the ball kept low.
The Amla-du Plessis stand was the second highest for South Africa in this series, after the 76-run partnership between Amla and Dean Elgar for the third wicket in the first innings at Mohali.
The pitch seemed to play a little slower than on the first two days, and Amla and du Plessis negotiated the bowling with some difficulty, as the ball found the edge of the bat a few times but did not carry to the fielders.
The pair looked more confident in the second session, but the runs did not come easy as they scored only 21 off 19 overs in the first hour after lunch.
It was only a matter of time once Amla and du Plessis departed, and Ashwin was back to finding the right spots on the pitch. The lower-order batsmen tottered amid mounting pressure, and India claimed only a third series win over South Africa in 12 overall.
Earlier, resuming on 32-2, South Africa lost two early wickets before Amla and du Plessis forged their partnership.
Ashwin, who took five wickets in the first innings, had Elgar caught bat-pad at silly-point for 18 and AB de Villiers was beaten by a ball that trapped him lbw for 9.
The last test will be in New Delhi from Dec. 3-7.
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