England faltered after the good start by the openers to reach 182 for 3 at lunch in their second innings after following on, even as the home team pressed hard for victory.
The hosts had scored a huge first innings total of 521 for eight declared before bundling England all out for 191 in their first essay.
Cook, overnight 74 in team's second innings total of 111 for no loss, remained unbeaten on 109 at lunch today at the Sardar Patel Stadium. The captain exhibited patience and skillful batting on a turning track.
Ian Bell who was on 11 at the interval was sent back by Umesh Yadav for 22 ending the 39 run partnership with the skipper.
Umesh struck again off the very next ball when he had Samit Patel LBW for a blob.
Pragyan Ojha, who had picked up five for 45 to help skittle out England for a paltry 191 in the first innings, was again in focus with impressive figures of two for 54 in 23 overs (till lunch).
The left-arm spinner, brought on quite late into the attack, accounted for two wickets in a space of seven balls, by packing off Jonathan Trott for 17 and star batsman Kevin Pietersen for 2 to leave England tottering at 160 for 3 half an hour before lunch.
Senior left arm pacer Zaheer Khan gave India the initial breakthrough after 28 minutes into the fourth morning by dismissing debutant opener Nick Compton for 37.
Compton was lucky not to be given out when he was rapped in front by the unlucky Ravichandran Ashwin and then survived a clear stumping chance by Dhoni off the same bowler.
Cook and Ian Bell, however, prevented further damage after the fall of Trott and Pietersen in quick succession.
Zaheer took one for nine in seven overs, while Ashwin, who had claimed 3 for 80 in the first innings, was wicket-less after bowling 11 overs this morning.
England progressed slowly this morning after commencing at the overnight 111 for no loss, still needing 219 runs to wipe off the 330-run first innings lead taken by India.
Cook was his usual calm self but the other opener, Compton, looked a bit ruffled despite seeing his left-handed partner square cutting Zaheer's short ball for four in the first over.
Compton, 34 last evening, was clearly lucky to escape a leg before wicket appeal off Ashwin who started the bowling with Zaheer, umpire Aleem Dar turning down the vociferous shout from the Indians.
The right hander was again fortunate when on 36 as wicketkeeper Dhoni missed an easy stumping chance off Ashwin who drew him well out of the crease.
However, his luck did not hold out for long as after the addition of one more run he was trapped leg before by Zaheer when he played across. In all, Compton faced 128 balls and hit two fours.
Trott, out for a duck in the first innings, looked better prepared this time as he cover drove Ojha for two fours. But the bowler switched to the end from where Ashwin bowled and struck two big blows in successive overs to send back Trott and Pietersen.
Pietersen, who was dismissed by Ojha in the first innings, again fell victim to the same bowler.
Cook reached his 100 in the 23rd over. The visiting team captain's 21st century in his 84th match was also his third against India following his unbeaten second innings century on his Test debut at Nagpur six years ago and his double hundred at Edgbaston in the last series between the two teams in 2011.
Cook also became the eighth English batsman to score more than one hundred on the Indian soil.