Rahane struck a cultured 127 and featured in a series-high 98-run partnership with Ravichandran Ashwin (56) before the hosts folded in the fifth over after lunch.
South Africa began cautiously before paceman Umesh Yadav drew first blood, getting Dean Elgar (17) caught behind in glorious sunshine at the Ferozeshah Kotla Stadium.
Temba Bavuma (20) and skipper Hashim Amla (one) were in the middle at the break with South Africa trailing India, who have taken an unassailable 2-0 lead in the series, by 296 runs.
India resumed day two on 231-7 and Rahane, on 89 overnight, stitched a third 50-plus partnership in the match with Ashwin before bringing up his fifth test century with an immaculate straight drive past fast bowler Kyle Abbott.
An industrious batsman in a lineup teeming with strokeplayers, the 27-year-old raised his arms and looked skywards in a mixed sense of relief and elation, having failed to go past 15 in the ongoing series.
Celebration followed with Rahane launching off-spinner Dane Piedt over mid-wicket and long on for two sixes in three deliveries.
The number five fell to a soft dismissal, driving Tahir to AB de Villiers at cover to depart after crafting a stellar knock over five hours with the help of four sixes and 11 boundaries.
Ashwin, currently the leading wicket-taker in the series, hit Tahir for a six to bring up his sixth test fifty to prove he was no dud with the bat either.
Abott dismissed Ashwin and Ishant Sharma in three deliveries to finish with 5-40 but the visitors could not stop India from posting the first 300-plus score in the series. – ReutersSouth Africa were 38-1 at tea on the second day of the fourth and final Test, losing opener Dean Elgar shortly before the break.
Original source: Proteas lose Elgar cheaply