Mohammed Shami took five wickets as India overcame late South Africa resistance on the final day to win the first Test by 203 runs at Visakhapatnam.
Shami returned figures of 5-35, while Ravindra Jadeja (4-87) claimed three wickets in an over, as the Proteas were dismissed for 191 in pursuit of 395.
Senuran Muthusamy (49no) and Dane Piedt (56) offered some resistance for South Africa, recovering from 70-8 with a 91-run partnership for the ninth wicket to delay India's push for victory.
But, Shami returned to wrap up the innings, taking the final two wickets to fall to clinch his five-for and earn the hosts a 1-0 lead in the three-Test series..
With South Africa starting the day 11-1 in the final innings, Ravichandran Ashwin had kicked things off for India, bowling Theunis De Bruyn for 10 to join former Sri Lanka spinner Muttiah Muralitharan as the quickest to claim 350 Test wickets.
Shami then bowled all three of Temba Bavuma, Faf Du Plessis and Quinton De Kock, with Jadeja taking care of Aiden Markram, Vernon Philander and Keshav Maharaj in the 27th over as the tourists crumbled.
India were held up by Muthusamy and Piedt in the afternoon, but they couldn't quite take the game beyond tea and into a final session.
Earlier in the Test match, Rohit Sharma registered twin hundreds in his maiden game as a Test opener. The two teams meet again in the second Test in Pune, starting on Thursday.