Chasing 212 in tough conditions at Newlands, South Africa were 60/1 in the 21st over when Elgar got the reprieve, but with the Indian camp busy with the DRS controversy the hosts went on to score 40 runs in the next eight overs.
The brittle Indian batting line-up meant the visitors failed to conquer South Africa for the eighth successive tour spread over a little over 29 years.
The Indian players were upset by the DRS decision that ruled in favour of South African skipper Dean Elgar.
Rabada scalped six wickets, three in each innings, to play a pivotal role in South Africa's series-levelling seven-wicket win in the second Test.
South African batters dug their heels in to reach within 122 runs of a series-levelling victory against a determined India as the second Test seemed poised for a gripping fourth day finish.
South Africa knocked off a target of 67 runs without loss as openers Dean Elgar (31 not out) and Aiden Markram (36 not out) saw them to victory at the Wanderers with two-and-a-half days of the contest still remaining.
If the Proteas dominated the first two sessions of the day, the last session was clearly shared as the spin duo of Ashwin and Ravindra Jadeja clawed their way back.