The hosts, 2-1 up in the series, lost young opener Sam Robson for 37 in the second over before Cook and Gary Ballance shared an unbroken second-wicket partnership of 82 to turn the screw on India who were skittled out for 148 on the opening day.
Cook, emerging from a wretched run of form, was 56 not out at the interval with Ballance on 40.
Robson missed a full straight ball from Arun Aaron which crashed into his off stump to end a first-wicket stand of 66.
Ballance and Cook played comfortably and accumulated runs steadily with crisp strokes off their legs and the odd fluent drive.
Ishant Sharma beat the bat a few times in a hostile spell of pace bowling from the Pavilion End but the Indians looked flat in the field after losing the last two tests heavily.
Cook reached his fifty with a square cut for four off spinner Ravichandran Ashwin, his fifth boundary, and England have a solid platform to build a commanding lead as they bid to clinch the series.