Resuming on 77 in reply to India's mammoth first-innings 587 all out, England lost Joe Root and Ben Stokes to successive balls from Mohammed Siraj in the second over but recovered brilliantly thanks to a 303-run stand by Brook (158) and Smith (a career-best 184 not out).
After taking the second new ball, India claimed England's last five wickets for 20 runs in just 44 balls to dismiss the home team for 407 - Brook and Smith accounting for 342 of those runs - and take a lead of 180 into the second innings.
England had a remarkable six ducks in total and Siraj returned figures of 6-70.
Batting under cloud cover, the Indians reached stumps on 1-64, with only the loss of Yashasvi Jaiswal (28), and will resume 244 runs ahead with a series-levelling victory in their sights.
Lokesh Rahul was unbeaten on 28 alongside Karun Nair, on 7.
England were in a big hole when Root and Stokes departed to leave their side on 5-84 and still trailing by more than 500 runs.
Siraj dismissed England's best batter - Root for 22 - by enticing a nick down the leg side, with wicketkeeper Rishabh Pant taking a diving catch, and then their captain for a golden duck.
Stokes trudged back to the dressing room after a steeply rising and seaming delivery came off the shoulder of his bat and flashed to Pant.
England were rocking, but Smith and Brook played as if there was no pressure with a counterattacking riposte.
Smith came in to face a hat-trick ball, struck that for four, and went on to make the biggest score by an England wicketkeeper, surpassing Alec Stewart.
He raced to his second test hundred before lunch in just 80 balls - the third fastest by an Englishman - and including an over when he pulled and smashed Prasidh Krishna for a six and four fours.
Brook compiled his ninth Test hundred in 27 matches, getting to three figures after being dismissed for 99 in the first Test won by England at Headingley last week.
Brook, a star for England in all formats, reined in some slight frustration at India changing tactics and bowling wide outside off-stump to a lopsided field and looked good after going down with cramp.
Soon afterward, he was bowled by Akash Deep (4-88) and was serenaded as he walked off gingerly.
That precipitated England's late-order collapse that saw Brydon Carse, Josh Tongue and Shaoib Bashir all being removed for ducks, joining Ben Duckett, Ollie Pope and Stokes.
Tongue responded by trapping Jaiswal lbw but India finished the day in the driving seat.
