The Proteas, needing 218 to win, never recovered after a disastrous start left them 10-3 and were bowled out for 109.
Slow left-armer Jadeja took 5-21 on his comeback after being dropped for recent tours of Bangladesh and Sri Lanka.
India began the day on 125-2 in their second innings but were skittled for 200, with Cheteshwar Pujara hitting 77.
South Africa had won the Twenty20 series 2-0 and taken the five-match one-day series 3-2 to raise hopes of a first Test series win on Indian soil in 15 years - and gave themselves a chance by bowling India out cheaply for a second time.
The Proteas' bowling attack was lacking injured paceman Dale Steyn, off the field with a groin strain, but spinners Imran Tahir (4-48) and Simon Harmer (4-61) did the damage. Indeed, 34 of the 40 wickets in the Test fell to spin.
But South Africa, who trailed by 17 runs on first innings, failed to come to grips with the conditions. Their batsmen were rattled with the ball turning viciously and also keeping low on the odd occasion.
Stiaan van Zyl top-scored for the tourists with 36, but seven batsmen failed to reach double figures.
Jadeja and fellow spinner Ravichandran Ashwin took the new ball and finished with match figures of 8-94 and 8-90 respectively.
With Virat Kohli having secured his first home Test win as India captain, the second of the four-match Test series begins in Bangalore on 14 November.
Spinner Ravindra Jadeja took his third five-wicket Test haul as India beat South Africa by 108 runs inside three days in the first Test in Mohali.