Fast-medium bowler Kumar did the bulk of the damage with an impressive return of four for 46 in 23 overs, his haul including all of the top four England batsmen as the home side reached stumps on 219 for six. (Have Learnt From England Bowlers, Says Bhuvneshwar Kumar)
That left them 76 runs behind India's first innings 295 after Cook had fallen for just 10.
Only Gary Ballance (110) and Moeen Ali (32) offered much in the way of resistance in a fifth-wicket stand of 98 after England, despite an easing pitch and a sun-drenched day making conditions for batting far less difficult than on Thursday, slumped to 113 for four.
Cook was caught behind by opposing skipper Mahendra Singh Dhoni off Kumar who has looked at home in England despite playing all his six Tests before this series in India.
Left-handed opener Cook's exit meant it was now 26 innings since he had scored the last of his England record 25 Test hundreds, with his tally for the calendar year standing at a mere 107 runs in eight innings.
The 29-year-old's poor run of form has also coincided with an England slump that has seen the team go nine Tests without a win.
For most of his 29-ball innings spanning 40 minutes, Cook looked assured.
But, not moving his feet, Cook nicked a good length Kumar ball that swung away through to wicketkeeper Dhoni.
Australia-born opener Sam Robson, dropped in the slips on seven, squandered his reprieve by driving loosely at a wide Kumar ball and was caught behind having made 17.
Since the third of his three Ashes centuries at Chester-le-Street last year, Ian Bell had completed only four fifties and no hundreds in 17 Test innings prior to Friday's knock.
Bell had made just 16 when, trying to evade a short ball from Kumar, he gloved a gentle catch to third slip.
Joe Root, fresh his 154 not out in the drawn first Test at Trent Bridge, came in facing yet another salvage mission.
The 23-year-old's last two Tests at Lord's had yielded scores of 180 against Australia in 2013 and 200 not out against Sri Lanka last month.
But Root was lbw for 13 to left-arm spinner Ravindra Jadeja on Friday.
Ballance was fortunate when, on 32, he edged medium-pacer Stuart Binny between Dhoni and first slip for what was a keeper's catch.
But he made the most of his luck in a composed display that saw him accelerate to a hundred after an understandably watchful effort early in his innings.
The Zimbabwe born-left hander drove and cut Binny for two fours in as many balls to go to 93 before two more fours off the same bowler, a clip off his legs and an on-drive, saw Ballance to a hundred in 186 balls including 14 fours.
It was the 24-year-old's second hundred in five Tests and second in as many at Lord's after the Yorkshire batsman's 104 not out against Sri Lanka last month.
Ali's painstaking two-hour knock came to an end when he was lbw to a fellow part-time spinner in Murali Vijay.
Ballance followed soon afterwards when, with India having taken the new ball, he was well caught down the legside by Dhoni off Kumar to end a five-hour stay at the crease.
India had been in dire straits at 145 for seven.
But England let them off the hook as India doubled their total to 290 for nine at Thursday's close, with Ajinkya Rahane making a superb 103 in his maiden Test innings at Lord's.