At 509/6, and 193 runs ahead of the Indians, England seem to be in a position where they cannot lose the match, while the hosts need to gear up for a grim fight for survival. With the four-match series now locked at 1-1, the winner at Eden Gardens will take an unassailable 2-1 lead in the four-match series.
Matt Prior (40) and Graeme Swann (21), who so far have stitched together 56 runs, were at the crease at the close of play.
Cook (190) fell ten short of his third double hundred when he was run out for the first time in Tests, and that too in a bizarre manner. The left hander took evasive action and the ball hit the stumps while he still was away from the crease. The incident brought some cheers for the home crowd who had nothing much to applause till the dismissal.
Buoyed by the return of Cook - the highest run getter in the series so far - the Indians bowled well and tasted success, taking three English wickets.
Ian Bell (5) edged Ishant Sharma behind the wicket, Mumbai Test Man of the Match Kevin Pietersen (54) was trapped in front by offie Ravichandran Ashwin after completing a half century, while Virender Sehwag juggled before latching on to a thick edge offered by Samit Patel (33) off Pragyan Ojha.
Ojha (3/140) was the most successful Indian bowler while Sharma and Ashwin chipped in with a wicket each.
Earlier, resuming at 216/1 overnight, Cook and Jonathan Trott (87) went from strength to strength crossing personal milestones. They remained unseparated in the morning session as England reached 296/1 at lunch.
Cook notched up his 150 in the 89th over as England went past 250 runs. Trott hit consecutive fours off pacer Zaheer Khan on way to completing his 13th Test 50 in 63 innings.
Luck smiled on Cook - then on 156 - when he lobbed a simple catch back to bowler Sharma, who could not hold on to the opportunity. This was the second reprieve for the southpaw as he was dropped at first slip by Cheteswar Pujara on 17.
Dhoni took the second new ball in the 89th over but the Indian bowlers tasted success only in the 120th when Ojha had Trott caught behind.
Cook and Trott put on a 173-run partnership for the second wicket - the second century-plus stand of the innings after the 165-run stand between the England captain and fellow opener Nick Compton.
Cook, however, continued to grind runs, while hitting the occasional boundaries, and even clobbered Ashwin for a six down the ground demonstrating England's urgency to step up the scoring rate.
With the wear and tear on the pitch evident, Dhoni started the post lunch session with his spin duo of Ojha and Ashwin who immediately settled into a good rhythm troubling the batsmen as the wicket afforded turn.
Meanwhile, an unfazed Cook went past 500 runs in the series, only the third batsman from his country to do so in a series in India after Ken Barrington and Mike Gatting.
With Indians showing positive intent, they were immediately rewarded with some good fortune with Cook departing just 10 short of a double century.
Along with West Indian great Vivian Richards and India opener Sehwag, Cook is one of the three batsmen to get out for scores in the 290s, 190s, 90s in Tests.
Mumbai Indians won the toss and decided to bat against Sunrisers Hyderabad in their Indian Premier League match here Wednesday.