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.
"In T20Is and ODIs, the bowlers get tested all the time".
India, who are facing a huge deficit of 272 runs, were 17 for two in their second innings when rain forced an early lunch.
"It's still a bit of a blur - incredible feeling", said Woakes, who hit 18 boundaries and took 2-19 in India's first innings of 107.
In the second test match between England vs India at Lord's cricket stadium in London, Virat Kohli and company will look forward to bounce back on track.
And to what became the last ball before lunch, Root was lbw for 19 to a off-cutter that kept low, with Bairstow then unbeaten on four.
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Inbetween Shami's strikes, Ishant Sharma had Alastair Cook (21) caught behind and Hardik Pandya (2-66) removed England debutant Ollie Pope for 28 to help bring India back into the match.
With Woakes, the first of England's three bowling all-rounders, at the crease alongside Bairstow, India could sense an opportunity to expose the tail.
India had recalled left-arm wrist-spinner Kuldeep Yadav, who tormented England during the preceding one-day series, for just his third Test after dropping paceman Umesh Yadav, who bowled well at Edgbaston.
Enter Kohli, but he was to lose Pujara after 87 balls of admirable defiance when Broad bowled him off his pads with a ideal inswinger.
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.
Sam Curran struck 22 not out from 24 balls as England chose to continue batting despite Sunday's forecast predicting rain. But Bairstow, as he's often done before, held firm while attacking the loose ball.
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