The opener batted throughout day four to reach 168 not out, taking the tourists to 340-5 in their second innings – a lead of 10.
England looked to be heading for defeat when Ian Bell and Samit Patel fell to successive Umesh Yadav deliveries. But Matt Prior (84 not out) arrived to add an unbroken 141 with Cook, raising English hopes of an unlikely draw.
The efforts of the sixth-wicket pair, the only two England batsmen to get to grips with the India attack on a slow track, denied the hosts what could have been an innings victory.
They came together with England 199-5 – 131 short of making the home side bat again – and first negotiated the reverse-swinging old ball, then saw off the new, before taking England into the lead.
Cook, who survived a strong lbw shout late on day three, did not offer a chance in his determined bid to ensure that his first Test since being appointed Andrew Strauss' successor does not end in defeat.