Cook, who was batting on 182 not out, went ahead of David Gower, Michael Vaughan and Kevin Pietersen in England's all-time list of centurions to go level with Len Hutton and his captain Andrew Strauss, who missed a great chance to score his 20th Test ton here at the Edgbaston.
Cook and Strauss (87) started the day at 84 for no loss and went on to score a 186-run opening stand, a record against India. At stumps, England stretched their lead to 232 and giving company Cook was Eoin Morgan, who was batting on 44.
It was a hard day for the Indian bowlers as England plundered 372 runs in the day after a delayed start due to rain. What made worse for India were the dropped catches with Rahul Dravid spilling two easy chances at the slips.
The most impressive bowler was medium pacer Praveen Kumar, who picked up two wickets, including that of the in-form Ian Bell (34) whose off-stump went flying.
Barring Kumar, none of the Indian bowlers could make an impact as Strauss (87) and Cook continued from where they left Wednesday evening to put England, who lead the four-match series 2-0, firmly on course for a big total.
Strauss and Cook began with an intent to occupy the crease and see off the initial overs. They played with soft hands and adjusted well to the seaming ball. The first boundary, however, came only in the 15th over of the morning. They slowly gained in confidence and freed their arms as England took lunch at 157 for no loss.
Strauss and Bell were the two wickets to fall in the second session but that did not impact the run flow. England added 162 runs in the second session in 35 overs at a brisk pace of 4.2 per over.
Strauss, who batted solidly, was unlucky to miss his century. He was bowled while sweeping Amit Mishra. Replays later showed that the leg spinner had overstepped but was not called by the umpire.
The wicket, however, came a big relief to the Indians as Strauss and Cook added 186 runs for the opening stand to push the visitors on the backfoot. Strauss hit 13 fours in his 176-ball innings.
Bell got a reprieve when he was on 30 when Dravid dropped an easy catch in the slips off Sreesanth. The local boy, however, did not hang in for long and Kumar, who had bowled beautifully in the morning, got rewards as he produced a stunning leg cutter that sent the off stump cart-wheeling.
Cook and Kevin Pietersen (63) then added 122 runs for the third wicket to frustrate the Indians. Left-handed Cook, who went through a brief dry spell, batted himself to form. It was his 19th Test century and sixth since the Ashes.
Cook had looked uncomfortable at the start of the innings, but played patiently and applied himself to grind it out in the middle.
Pietersen was also in fluent touch, hitting well-timed nine fours and a six, before being trapped leg-before by Kumar. Morgan made a shaky start but slowly grew in confidence.
The left-handed batsman got a life in the final over off Mishra as Dravid messed up another chance to sum up the hard day for the visitors.
