England resumed on 357 for six, already 250 runs ahead of India's meagre first-innings 107 all out.
The seventh-wicket pair Hardik Pandya and Ravi Ashwin kept England waiting after tea, with a stand of 55, but Woakes quickly parted them to put his team on the home straight.
Broad (four for 44) and Anderson (four for 23) each recorded a notable new milestone as England bowled India out for 130 on the fourth day. By contrast, Saturday " s play started under sunny blue skies that promised to make life easier for the batsmen.
In 17 Tests since Joe Root became captain, England have been dismissed for under 300 in their first innings seven times.More news: How blue light accelerates blindness
Earlier, England's top order again failed with openers Keaton Jennings (11) and Alastair Cook (21) falling in successive overs.
The exit of England´s all-time leading Test run-scorer meant 20-year-old debutant batsman Ollie Pope walked out with the hosts in trouble at 32 for two. It took the Surrey right-hander just two balls to score his first Test runs when he flicked a Sharma inswinger behind square-leg for four.
Pope made a promising 28 before he was lbw to Pandya.
Jos Buttler was the one wicket to fall in the afternoon session, trapped lbw for a breezy 24 by the impressive Mohammed Shami who claimed three wickets.More news: Nobel Prize-winning author V.S. Naipaul is dead
Bairstow received sound support from the correct Woakes, selected after fellow all-rounder Ben Stokes was omitted because of clash with his ongoing trial for affray.
A quick word from Bairstow helped Woakes to compose himself and a pull shot over mid-wicket for three took him to the landmark off just 129 balls.
India's decision to replace pacer Umesh Yadav with slow left-arm bowler Kuldeep Yadav came into question as the tourists struggled to maintain control on a wicket that appeared better suited to seamers than spinners.
Bairstow hit eight boundaries and scored freely to keep the pressure on the India bowlers and Woakes also struck eight fours following his two wickets on Friday.More news: Trump condemns racism, year after Charlottesville
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