On a near-perfect day for the hosts at the Ageas Bowl, England were impressive with both bat and ball to leave India on 112-4 at stumps with a mountain to climb.
Birthday boy James Anderson (5-53) claimed the wickets of MS Dhoni (50) and Mohammed Shami (5) as India were bowled out for 330 early on.
Captain Alastair Cook's resurgence then continued with an unbeaten 70 to help England make 205-4 by tea before they put India back into bat.
The hosts' bowlers soon got to work again in the final session, as Mooen Ali (2-33) and Joe Root (1-5) struck after Muarli Vijay had been run out, to leave India requiring 333 more runs on the final day.
Dhoni and co must now make history and record the highest fourth-innings run chase if they are to win - a feat that would better West Indies' 418-7 achieved against Australia in 2003.
Starting the day on 323-8, India saw Dhoni and Shami dismissed inside five overs before Cook spared the tourists of the follow-on.
Seven overs into England's second innings, Stuart Robson (13) was caught by Shikhar Dhawan at slip off the bowling of Bhuvneshwar Kumar (1-59).
Gary Ballance made 38 before being given out just before lunch - Marais Erasmus judging Ravindra Jadeja's delivery had nicked off the batsman's glove on its way to Cheteshwar Pujara.
Ian Bell (23) was tempted into sweeping Jadeja before the ball ducked under his bat and hit the stumps while Cook moved past 50 as he and Root edged England to a 400-run lead with seven wickets remaining.
The pair put on a partnership of 99 runs before Root went for 56 and England declared on 205-4.
With 445 required for India to win, fine fielding from Stuart Broad and Chris Jordan accounted for Vijay (12) and Pujara (2) to leave the tourists 29-2 in the 13th over.
Pujara was Ali's third wicket of the Test and it perhaps should have been four soon after - his lbw appeals against Dhawan turned down despite the ball appearing to be hitting the stumps.
That was only a temporary reprieve, though, as Root and Ali struck shortly afterwards to remove Dhawan and Kohli respectively, before Anderson dropped Rohit Sharma low to his left as India ended the day on 112-4.