The crushing win on the last day of the Test saw the hosts wrap up the series 3-0 after having won the first and third matches. The rain-hit second game in Bangalore ended in a draw.
India had to work hard for Monday's win after the top-ranked tourists resorted to some stubborn defensive batting, scoring at barely a run an over in the hope of salvaging a draw.
But their plans were dashed by the equally determined Indian bowlers led by off-spinner Ashwin whose match haul of 7-87 saw him finish the series with a total of 31 wickets.
Refusing to lie down: South Africa stonewall their way through day four
Paceman Umesh Yadav chipped in with three wickets as South Africa were bowled out for 143 soon after tea, chasing a massive target of 481.
India declared their innings at 267-5 after making 334 in their first outing after South Africa folded up for 121 in their first knock.
South Africa fought bravely but their innings soon unravelled, losing five wickets in the space of just 31 balls in the final session.
AB de Villiers (43 off 297 balls) and skipper Hashim Amla (25) stonewalled the attack during their marathon innings but the other batsmen caved in without making much effort.
Rahane steers India to 231-7 against South Africa
De Villiers, who holds the record for the fastest half-century and century in one-day cricket, anchored one end and showed remarkable resolve on a slow track that offered occasional bounce.
The batsman appeared undeterred despite taking three painful blows on his hands against paceman Umesh Yadav.
De Villiers was finally removed by Ashwin who had him caught at leg slip.
Ravindra Jadeja (2-26) trapped Faf du Plessis (10) leg before wicket in the post-lunch session but not before he had played out 97 balls.
Ashwin spins India to series win over S.Africa
Amla and de Villiers had resumed the day with the same dogged resolution that they showed on Sunday.
Their ultra-defensive batting clearly frustrated Indian skipper Virat Kohli, who even brought himself on to bowl one over.
But it was Jadeja who gave his team the key breakthrough with the new ball.
The left-arm spinner made the ball drift into the right-handed Amla. He tried to defend once again but the ball sneaked past to hit the top of his off-stump.
Amla's nearly five-hour stay at the wicket yielded just 25 runs off 244 balls, the slowest in Tests by a batsman who has played 200 balls or more.
The 27-run stand that Amla shared with de Villiers off 253 balls was the lowest among all Test partnerships involving 200 or more balls.
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