Having scored 141 runs in the afternoon session to take the lead past 100, England continued to dominant after tea, with Bairstow reaching his half-century and Woakes following suit shortly after.
When bad light forced an early close on the third day, England were 357 for six in reply to India´s meagre first-innings 107 - a commanding lead of 250 runs.
England prefer Chris Woakes to Moeen Ali as the replacement for Ben Stokes, while Ollie Pope is in for Dawid Malan.
In the first over back, Kohli, off strike, refused an easy single called by Pujara.
And in his 25th match at this level, Woakes surpassed his previous Test best of 66, made against Sri Lanka at Lord´s two years ago, before completing a 129-ball hundred, including 15 fours, with a pulled three off Hardik Pandya.
"It´s hard to put into words, I suppose" Woakes told Sky Sports. He raced to his first Test century and at stumps was 120 not out. "To receive a standing ovation at Lord's is a bit of a boyhood dream". I don't know whether it's the surface or the conditions or I've just hit form at the right time.
Woakes surged past Bairstow and was suddenly in sight of securing a place on the batting honors board at Lord's - to go along with his place on the bowling honors board after taking two five-wicket hauls at Lord's against Pakistan two years ago. With with very few runs by the batsmen in the first innings, it was a daunting task for the Indian bowlers to prevent the home team from going anything big in their first innings.
India had lost the first Test by 31 runs in Birmingham where they failed to chase down a 194-run target set by England. Five balls later Jennings " s fellow left-handed opener Alastair Cook (21) was out as well, caught behind off a superb Sharma delivery from around the wicket that squared him up and seamed away before finding the outside edge.
Jos Buttler (24) got in and got out before he too fell lbw aiming across the line as Shami took a third wicket this innings.
Earlier, India chose to play with two spinners in Ravichandran Ashwin and Kuldeep Yadav, whereas Cheteshwar Pujara had a horrific return to Test cricket.