Irfan Pathan (52), Harbhajan Singh (36) and Anil Kumble (32) all built on Dravid's knock as India were bowled out for 338 in their first innings in reply to England's 300.
Tea was taken immediately after the end of India's innings.
India were in danger of conceding a big lead in the morning session when they were reduced to 153-5 after resuming on 149-4, but Dravid and Pathan steadied the innings with a vital 76-run stand for the sixth wicket.
Kumble, who became the fifth bowler in Test history to complete 500 wickets on Saturday, and Harbhajan were then involved in a rollicking 53-run stand for the eight wicket to help their team gain the lead.
Stand-in England captain Andrew Flintoff was the most successful bowler, claiming three of the six Indian wickets that fell on Sunday to finish with 4-96. Swing bowler Matthew Hoggard and paceman Steve Harmison each bagged two.
Flintoff sustained his team's hopes of gaining the lead when he dismissed well-set Dravid and the free-stroking Pathan in the morning session.
Dravid looked set to complete his 23rd Test century before he inside-edged a delivery on to his stumps. He added 35 to his overnight score of 60 and hit 10 fours, including three in an over from seamer Liam Plunkett.
Pathan struck one six off left-arm spinner Monty Panesar and five fours in his sixth Test half-century before being caught by Paul Collingwood at backward point off Flintoff.
England then faced stiff resistance from unexpected quarters as tail-ender Harbhajan chanced his arm to smash seven fours in an entertaining 41-ball knock.
Last-man Munaf Patel also played a couple of big shots, including a straight six off Hoggard.
Harmison had looked the most impressive bowler in the first session, dismissing hard-hitting Mahendra Dhoni and then testing both Dravid and Pathan in his sharp six-over opening spell.
Dhoni hit Harmison for a four through the covers before being surprised by extra bounce in the same over, the ball brushing the batsman's gloves on the way to wicket-keeper Geraint Jones. He added four to his overnight 12.