Left-arm spinner Ravindra Jadeja grabbed five wickets and off-spinner Ravichandran Ashwin took three as the Proteas, set a victory target of 218, were shot out for 109 after tea on the third day.
Both bowlers finished with eight wickets in the low-scoring match played on a dusty, dry pitch at the I.S. Bindra stadium, with the Indian spinners claiming 19 of the 20 South African scalps.
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The Proteas were reduced to 60-6 before Stiaan van Zyl, who top-scored with 36, and Simon Harmer (11) got together to put on 42 for the seventh wicket.
Seven batsmen failed to reach double figures, leaving the top-ranked South Africans a lot to ponder ahead of the second Test in the four-match series which starts in Bangalore on November 14.
Earlier, South African spinners Harmer and Imran Tahir claimed four wickets each to rip through India's batting in the morning session.
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India, who started the third day at 125-2 in their second innings, moved to 161-2 before a batting meltdown saw them lose their last eight wickets for 39 runs.
South Africa fared worse when they batted a second time on a wicket that has proved to be a batsman's nightmare.
The gamble to open the batting with tailender Vernon Philander backfired when the fast bowler was leg-before to Jadeja for one in the second over.
Ashwin, who had taken five wickets in the first innings, struck in the third over as Faf du Plessis edged an easy catch to Ajinkya Rahane in the slips.
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From 9-2, the Proteas became 32-4 as Jadeja bowled skipper Hashim Amla for no score and star batsman AB de Villiers (16) also found his stumps shattered by leg-spinner Amit Mishra.
The wickets continued to tumble as opener Dean Elgar (16) top-edged an intended pull off seamer Varun Aaron and spooned an easy catch to Indian captain Virat Kohli at mid-on.
In the morning, the loss of three top-order batsmen in the space of three runs triggered India's collapse even though the Proteas fielded without their injured pace spearhead Dale Steyn.
The overnight pair of Cheteshwar Pujara and Kohli batted through the first hour to take their third wicket stand to 66 runs.
Medium-pacer van Zyl, the sixth bowler summoned by Amla, broke the threatening partnership soon after the drinks interval by having Kohli caught behind for 29.
Pujara himself fell two overs later for 77, edging leg-spinner Tahir's first delivery to Amla in the slips.
South Africa gained another wicket in the next over when Rahane was caught smartly at short-leg by substitute fielder Temba Bavuma off Harmer.
The off-spinner earned two more quick wickets to reduce India to 182-7 as he trapped Jadeja leg-before for eight and had Amit Mishra caught at backward square-leg for two.
Tahir had Ashwin caught in the slips off the last ball before lunch and then terminated the innings after the break by having Wriddhiman Saha caught behind for 20.
The tourists took the field without Steyn, who was sidelined for the entire innings due to a groin strain.
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