The visitors were all out for 191 in the second session after resuming on 11 on the day, chasing an improbable 395 for victory.
In the second innings, after Rohit Sharma registered twin hundreds as a test opener in his maiden match to set up India’s drive towards victory.
And the Indian bowlers have showcased some clinical display and with their early strikes, Shami snuffed out the hopes of South Africa to save a draw and later returned to complete his fifth five-wicket journey in Tests.
4-87 was picked up by Jadeja to help India take a 1-0 lead in the three-match series, including three in one over.
Debutant all-rounder Senuran Muthusamy and Tailender Dane Piedt showed the grit and determination missing from their top and middle-order batsmen, stitching a 91 wicket stand together to delay the victory of India.
Shami who took 5-35, was left to break the stand by dismissing Piedt for the fourth bowled-out victim after the batsman had finished trying his maiden for half a century.
31 wickets in four tests were taken by Ravichandran Ashwin in four tests to lead India to a 3-0 series won the last time South Africa traveled to India in 2015, got the early breakthrough on Sunday when Theunis de Bruyn hit the off-spinner to be 10.
Ashwin has 350 Test wickets in his bag and equals the record for the fastest in the world with Sri Lankan spinning great Muttiah Muralitharan who reached the milestone in his 66th match.
In the first innings with the spinners by taking nine out of 10 wickets on a turning pitch and there were a few deliveries which remained low, Shami demonstrated what it takes for a quick bowler to succeed on such a ground. He bowled quickly and hit the stumps and his three bowled-out dismissals were awarded with a wicket.
Even though the ball remained a low touch to knock Temba Bavuma’s timber, and the skipper Faf du Plessis was out for 13 to a horror verdict when he shouldered to a Shami delivery which came in and struck the off stump. As Shami split his cover, the first innings centurion Quinton de Kock was out for a second-ball duck.
Jadeja then stepped in and benefited from his precision on favorable ground. With an incredible one-headed return grab, he dismissed Markram, a then Vemon’s philander and Keshav Maharaj all in five deliveries to shock South Africa.
It was the third straight victory in as many matches in the world test championships for top-ranked side India. Starting Thursday at Pune, the second Test begins.