India seamers Jasprit Bumrah and Mohammed Shami took three wickets each as the tourists bowled South Africa out for 191 to claim victory by 113 runs just after lunch on the fifth day of the first Test in Centurion on Thursday.
South Africa had resumed on 94 for four chasing a venue record 305 for victory, but fell well short of their target on a spicy wicket that had plenty of assistance for the bowlers.
Dean Elgar (77) and Temba Bavuma (35 not out) gave them some hope in the morning session but India's vaunted pace attack used the conditions superbly and eased their side to a comfortable win.
The second Test will be played in Johannesburg from January 3-7, with the third set for Cape Town from January 11-15. India are seeking a first-ever Test series win in South Africa.
"We got off to the perfect start and getting a result within four days, with one day washed out, shows how well we played and how motivated we are," India captain Virat Kohli said at the post-match presentation.
"The discipline the batsmen showed ... we knew anything over 300 or 320 (in the first innings) would be a top total. We had a lot of belief in our bowling unit."
Kohli said the team backed their pace attack to bowl teams out in overseas tests and had special praise for Mohammed Shami, who took 8-107 in the match.
"He is a world class talent. He is easily among the best three seamers in the world right now. He seems to get more out of the pitch that anyone else and bowls in areas where you can't leave the ball."
South Africa went along well in the first 10 overs of the day before Elgar was trapped leg before wicket by Bumrah (3-50).
Quinton de Kock took the attack to the bowlers with a breezy 21 from 28 balls, but played onto his own stumps for the second time in the Test, on this occasion off Mohammed Siraj (2-47).
It is a shot he will want to forget and he will play no further part in the series as he leaves the squad on paternity leave.
It took India only two overs to take the final three wickets after the lunch interval as spinner Ravichandran Ashwin (2-18) finally found a way into the game.
"We did a few things wrong, but there are also a lot of positives to take out of this game and into the next few (tests)," Elgar said.
"We will have a sit-down with myself and the management. I don’t think we did a lot wrong, but the things we did do wrong we were quite far off.
"It’s not all doom and gloom for us, we have put ourselves under pressure but that is generally when we thrive."