Nair became the second Indian batsman to score a test triple hundred and his 303 not out helped the hosts post a massive 759 for seven before skipper Virat Kohli declared.
After conceding a 282-run first-innings lead, England finished on 12 for no loss, needing to bat out the final day to save the match and prevent a 4-0 series loss.
Skipper Alastair Cook was unbeaten on three at the close with Keaton Jennings on nine after a difficult day in the field for the touring side.
Nair struck his 32nd four off Adil Rashid to bring up his triple century before raising his bat in celebration as his coach Anil Kumble, a keen photographer, captured the moment on camera.
Nair was at the other end when Lokesh Rahul, his partner in a 161-run fourth-wicket stand, fell agonisingly short of his double century on Sunday by playing a rash shot when one run away from the milestone.
Playing only his third test, Nair brought up his maiden century in the morning session, turned it into a double after tea and accelerated to reach the 300-mark in an innings which also included four sixes.
Resuming on 391 for four, India lost Murali Vijay for 29 but Nair and Ravichandran Aswin added 181 runs for the fifth wicket to put the hosts firmly in command.
Batting down the order following a shoulder injury, Vijay became debutant Liam Dawson's first test victim when the left-arm spinner trapped him lbw.
Nair, dropped on 34 by Cook, enjoyed another slice of luck when he hung out his bat trying to leave a Stuart Broad delivery, the deflection running past slip to the boundary.
Ben Stokes bowled with hostility, generating reverse swing from an otherwise flat track, and nearly had Nair caught at mid-off with another short delivery.
Nair was on 217 when Joe Root dropped him at slip off Jake Ball and the batsman also survived a stumping opportunity.
Ashwin hit a six off Moeen Ali and registered his 10th test fifty before he fell for 67 to a brilliant diving catch by Jos Buttler off Broad.
Virender Sehwag is the only other Indian to make a test triple century, achieving the feat twice.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Ed Osmond)
