England were made to follow-on in Ahmedabad after being bowled out for 191 in reply to India's 521-8 on day three.
Left-arm spinner Pragyan Ojha finished with 5-45 as England slipped from 41-3 at the start of the day to being bowled out just before tea.
When Matt Prior was last out for 48 - the top score of the innings - India captain Mahendra Singh Dhoni had no hesitation in asking England to bat again with the tourists still 330 behind.
Cook and Kevin Pietersen had dropped anchor during the first hour of the morning against spinners Ojha and Ravichandran Ashwin (3-44).
They had extended their fourth-wicket stand to 39 when Pietersen was once again undone by left-arm spin, Ojha bowling him middle stump for 17 with one that did not turn.
Ojha made it two in two balls when Ian Bell holed out to Sachin Tendulkar at mid-off after coming down the pitch to his first delivery.
Cook, having battled his way to 41 from 109 balls, was next to go, edging a flighted delivery from Ashwin low to Virender Sehwag at slip.
And just to pile on the misery, Samit Patel (10) received a poor lbw decision after seam bowler Umesh Yadav (1-6) had been brought into the attack for the first time shortly before lunch, which was taken with England on 110-7.
Prior struck nine boundaries and received some support from Tim Bresnan (19) and Stuart Broad (25) in the afternoon before Ojha completed his fourth five-wicket haul in Tests.