At stumps, England were 456 for three for in reply to India's mediocre 224 - a lead of 232 runs. Cook's fellow left-hander Eoin Morgan was on 44 in a fourth-wicket stand of 82.
England were in complete control as they pursued a win, that would give them both an unbeatable 3-0 lead in the four-Test series and see them replace India at the top of the ICC's Test Championship table.
The 26-year-old's 19th Test century came in his 71st match, after he had managed just 20 runs in his previous four innings.
Together with fellow left-handed opener Andrew Strauss, Cook shared a first-wicket stand of 186 before the captain fell for 87 to leg-spinner Amit Mishra.
He then put on 122 with Kevin Pietersen, who made a dashing 63, including a straight driven six off Mishra.
England started the day's final session on 319 for two, a lead of 95 runs, with Cook 129 not out and Pietersen unbeaten on 36.
Pietersen completed his fifty in an Ishant Sharma over where he scored 14 runs, including three boundaries with the best a clip from outside off-stump through midwicket.
Gooch has always encouraged Cook, who led England's batting statistics in Australia with 766 runs at an average of 127.66, including three centuries during the decent 3-1 Ashes series win, to score "daddy" hundreds.
And his protege, whose only Test double century to do date was the 235 not out he made against Australia in Brisbane in November, did that by pulling fast bowler Shanthakumaran Sreesanth to wide long to go to 150 with his 23rd four.
It was the fifth time in Tests that Cook had reached that landmark and the fourth since March last year.
* Agence-France Presse
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