At lunch on the third day, India were 108 for three in reply to England's imposing 569 for seven declared, a deficit of 461 runs after Broad had dismissed Cheteshwar Pujara (24) and Murali Vijay (35).
Virat Kohli was 18 not out and Ajinkya Rahane 11 not out, with India needing a further 262 runs to avoid the follow-on.
England had seen Ian Bell (167) end a run of 20 innings without a Test hundred and Gary Ballance (156) make his Test-best score on Monday.
Meanwhile England debutant wicketkeeper Jos Buttler, who might have out for a duck and gave two other chances, rode his luck to complete a typically quickfire 85.
Things got even better for under-pressure England captain Alastair Cook, who had earlier ended a run of low scores with a morale-boosting 95, when he caught Shikhar Dhawan at first slip off James Anderson before Monday's close.
Following a minute's silence Tuesday to commemorate cricketers who were killed during the First World War, a hundred years on from the start of that conflict, India resumed on 25 for one.
Opener Vijay, averaging 90 in the series, was 11 not out and Pujara four not out.
England, 1-0 down in the series, and without a victory in their previous 10 Tests, knew they had to 'win' Tuesday's play if they were to have a realistic chance of drawing level in this match.
After the batsmen made a solid start against England, with bowlers appreciably quicker than those playing for India in the absence of the injured Ishant Sharma, the seven-wicket hero of the tourists' 95-run win in the second Test at Lord's, Broad made the breakthrough.
Pujara, trying to sway out of the way of a well-directed short ball, didn't withdraw his bat and gloves in time and gave Buttler a simple catch for his first Test dismissal.
Kohli, whose best score in four previous innings this series was only 25, drove Chris Jordan through the covers for four.
But Broad struck again when Vijay, trying to withdraw his bat, deflected the ball onto his stumps and was bowled.
Broad had taken two wickets for 10 runs in 22 balls and India, on a sunny day ideal for batting, were 88 for three.
England thought they had Rahane, who made a superb hundred at Lord's, caught by Buttler for eight off occasional off-spinner Moeen Ali.
It was an excellent legside take by Buttler and the Hot Spot replay confirmed a thin touch on the glove.
However, Australia umpire Rod Tucker ruled not out and with India objections meaning the Decision Review System is not being used this series, Rahane survived.
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