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The win will be especially pleasing for England for the way they stuck to their task on a flat fifth-day wicket, with Stokes's superb burst earning the all-rounder figures of 3-35.
The teams now move onto Cape Town for the second Test at Newlands from Jan. 3-7.
Philander was an important part of the Proteas pace troika, comprising Dale Steyn and Morne Morkel, both of whom have retired from Tests.
There are 40 points on offer and a 3-0 series whitewash, both of which India would be targetting against the hapless visitors.
South African captain Faf du Plessis conceded that India were deserving series winners and his side was found wanting in the contest.
South Africa were bowled out for 189 in their second innings shortly after tea after India enforced the follow-on at the start of the fourth day.
If the morning belonged to Shami's hostile spell and an acrobatic catch from Saha, the afternoon session was all about relentless pressure created by Ashwin and Jadeja.
Kohli completed his seventh double hundred in Test cricket, remaining unbeaten on 254, while Ravindra Jadeja (91) missed out on his second hundred by nine runs.
India lost Rohit Sharma cheaply during a difficult opening spell from South Africa's fast bowlers, but weathered the storm to reach 273/3 at stumps.
Mohammad Shami scalped a five-wicket haul while Ravindra Jadeja picked up four wickets as India thrashed Proteas on the fifth and final day of the opening Test.