Cook's unbeaten 182, from 339 balls with 26 fours, helped England reach 456-3 at stumps on the second day. Eoin Morgan was twice dropped before finishing on 44 not out.
"There's no way this game's won, or even close to being won," Cook said. "We've got to go out tomorrow and score a lot of runs because I don't think batting last is going to be easy on this wicket."
Earlier, Andrew Strauss was out for 87 after hitting 13 fours in 176 balls. Kevin Pietersen made 63 with one six and nine fours from 78 deliveries before he was lbw to Praveen Kumar, who claimed 2-75 with 12 maidens to be India's best bowler.
India compounded its woes by bowling 46 extras, with Amit Mishra the most culpable with eight no-balls.
"There are no excuses, to be honest," India batsman Gautam Gambhir said. "It's just a matter of concentration. At a crucial moment we ended up dropping those two catches against Morgan, but these things happen in cricket. I can't tell you why.
"There was no shortage of effort. Everyone gave 100 percent, but sometimes the rub of the green doesn't go your way."
England began the day on 84-0 after rain delayed the start of play by 30 minutes, but after the excitement of England bowling India out for 224 on the opening day, Thursday's morning session was far cagier.
Neither Strauss nor Cook looked in peak form, but while India's bowlers did at least restrict the run rate to 3.34 an over before lunch, they failed to put the batsmen under any serious pressure.
Kumar has at times appeared to be taking on England almost single-handedly in this series and he again provided the most meaningful resistance, bowling four consecutive maidens.
Yet while Kumar bridled with aggression - earning the admiration of a home crowd that chanted his name in the evening session - some of his teammates shivered in the field, often keeping their hands in their pockets until the moment of delivery.
The mediocrity of the fielding was typified when Kumar found Cook's edge and the ball bounced short of Virender Sehwag, who let the ball go through his hands.
In the 45th over, Cook brought up his 50, from 130 balls, with a glance through square leg for four.
Mishra made a nightmarish start to the afternoon session, bowling three no-balls in an over, although he was unlucky not to remove Cook when the batsman was dropped at leg slip by Gambhir.
Mishra then had an lbw appeal turned down in the 56th over, but his next delivery - which should have been another no-ball - bowled Strauss, who was attempting to sweep.
Bell made a risky start but started scoring rapidly, helped by some loose deliveries from Sharma.
Bell swatted Mishra through square leg for the four that took England into the lead, but on 30 he was dropped by Rahul Dravid - the man who has taken more catches than any other in test cricket - at first slip.
Cook reached his 100, from 213 balls, with a leg glance to Kumar, but the next delivery finally saw Kumar get his reward with a superb ball that ripped out Bell's off stump.
An aggressive Pietersen smashed Mishra back over his head for six in the 78th over and reached his 50 from 53 balls before surviving a referral for caught behind off Sreesanth on 62.
Pietersen added just one before he was lbw to Kumar, but Sreesanth then dropped a simple chance to catch Eoin Morgan, who was on 17, off Sharma at backward point.
India's day was summed up when Morgan was dropped again, by Dravid, in the final over off Suresh Raina.
