The hosts, who resumed the day on 60 without loss, reached 256 for three at tea, still 221 runs behind, with Rahul on 133 and Karun Nair on 19.
The hosts, who resumed the day on 60 without loss, reached 256 for three at tea, still 221 runs behind, with Rahul on 133 and Karun Nair on 19 at the crease in Chennai.
England's Ben Stokes and Stuart Broad worked hard on a benign track to send Cheteshwar Pujara (16) and skipper Virat Kohli (15) back to the pavilion.
It was the first time in the series that Kohli, who leads the batting chart with 655 runs, had been out for less than 40. He chipped a drive to extra cover off Broad.
Rahul, who recorded his fourth Test century in just his 12th match, then toned down his aggressive play to take India's reply further in partnership with Nair.
His 228-ball stay has included nine fours and two sixes, and the stand is worth 45 so far.
Overnight batsmen Parthiv Patel and Rahul made England's bowlers toil earlier in the day with a 152-run opening stand, before Moeen Ali's off-spin prised out Patel -- the only wicket of the morning session.
It was India's first century opening partnership in 32 innings even though wicketkeeper Patel was filling in for regular opener Murali Vijay, who has not yet batted due to a bruised shoulder.
Patel was finally undone when he played against the spin of a flighted and sharply turning delivery from Ali to spoon a simple catch to Jos Buttler at cover. He recorded a Test-best 71 off 112 balls with seven fours.
India hold an unassailable 3-0 lead in the five-Test series.