At lunch on day five, England were 211 for two with Cook unbeaten on 106 and Ben Stokes at the other end on six not out.
Sixty one overs were left in the final day’s play for the touring side to push for a victory in the five-match series.
The 31-year-old Cook continued his penchant for scoring big against India by bringing up his sixth hundred against the opponents, five of them while touring the country.
Resuming on his overnight score of 46, the left-hander looked unperturbed against the turning ball and was equally comfortable against India’s pacemen, who got the ball to reverse swing.
Cook got to his three-figure mark by pushing leg-spinner Amit Mishra for a single and has hit 10 boundaries in his knock so far.
Mishra broke England’s best opening stand in India with his third ball of the day when Hammed, on 82, hit a flight delivery straight back to the spinner.
England’s debutant opener struggled in the morning, repeatedly beaten by India off-spinner Ravichandran Ashwin outside his off-stump.
The 19-year-old also appeared jittery against fast bowlers Mohammed Shami and Umesh Yadav, twice edging them through the vacant second slip area to pick up boundaries.
Mishra also picked up Joe Root in his second over of the morning when the right-handed batsman went for a wild slog and got a top edge for wicketkeeper Wriddhiman Saha to pouch the skier.
England promoted the hard-hitting Stokes up the order in an intent to score quick runs and set a target for Indian to chase.
Original source: Cook hits ton to lift England’s lead to 260