Virat Kohli enforced the follow-on at the start of the penultimate day after the Proteas were bowled out for 275 in their first innings on Saturday in reply to 601-5 declared.
South Africa's shortcomings with the bat were exposed again on Sunday as they were dismissed for only 189, suffering their heaviest defeat to India in the longest format.
Umesh took 3-22 and Jadeja claimed 3-52, while Ravichandran Ashwin (2-45) also did damage as India sealed a record 11th consecutive series on home soil.
India will be out to secure a 3-0 whitewash when the third Test gets under way in Ranchi next Saturday.
India win by an innings and 137 runs, sealing the series against South Africa and maintaining their 100 per cent World Test Championship record
— ICC (@ICC) October 13, 2019
Ishant Sharma struck with only the second ball of the day after Kohli asked the tourists to bat again, Aiden Markram bagging a pair when he was snared leg-before.
Theunis de Bruyn fell cheaply to Umesh when Wriddhiman Saha took a superb one-handed catch and the wicketkeeper clung on again after a juggling act to send Faf du Plessis on his way for just five from 54 balls.
Dean Elgar took a more positive approach, but that proved to be opener's downfall as he became Ashwin's second victim, holing out to Umesh going for a ninth boundary two short of his half-century.
The Proteas were 79-5 early in the afternoon session, having lost three wickets for only nine runs when Quinton de Kock was bowled attempting to sweep Jadeja.
Temba Bavuma frankly admitted his ego had been dented after failing in the first innings, but the right-hander showed what he is capable of by pulling Jadeja for six and playing a few shots in a sixth-wicket stand of 46 with Senuran Muthusamy.
Jadeja had the last laugh as Ajinkya Rahane held on to a sharp chance to dismiss Bavuma for 38 and South Africa were 129-7 when Muthusamy fell to Mohammed Shami.
Vernon Philander dispatched Jadeja for an enormous six over long-on and the left-arm spinner was left frustrated when he would have had his fellow all-rounder lbw for 18 if Kohli had reviewed.
Philander and Keshav Maharaj put on 56 after also holding India up in the first innings, but Umesh returned to get the former caught behind for 37.
Kagiso Rabada departed in the same over after edging a hostile short ball to Rohit Sharma in the slips and South Africa were all out when Maharaj was struck in front by Jadeja.
Photo Getty Images Caption: Ravindra Jadeja