India lost seven second-innings wickets in the evening session as they were bowled out for just 131 in 34.1 overs.
Kohli and Shubman Gill (26) were the only India batters into double figures as Nandre Burger (4-33) and Marco Jansen (3-36) did the damage.
The second and final Test starts on 3 January at Newlands in Cape Town.
Virat Kohli's defiant 76 failed to save India from being thrashed by an innings and 32 runs by South Africa in the first Test at Centurion.
India lost seven second-innings wickets in the evening session as they were bowled out for just 131 in 34.1 overs.
Kohli and Shubman Gill (26) were the only India batters into double figures as Nandre Burger (4-33) and Marco Jansen (3-36) did the damage.
The second and final Test starts on 3 January at Newlands in Cape Town.
India were 62-3 at tea with Kagiso Rabada, who took 5-59 in the first innings, capturing the key wicket of India skipper Rohit.
However, the tourists lost seven wickets for just 59 runs in the evening session as South Africa's seamers ran through their middle order.
Kohli made a determined 76 from 82 balls but he was last man out - caught brilliant by Rabada off Jansen - as extras third top-scored for India with eight runs.
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Player of the match Elgar said: "You need fast bowlers and a spinner, potentially, to take 20 wickets. You need guys firing.
"Kagiso Rabada was amazing, and a guy like Nandre just came to the fore. It was a really great team performance."