England 290 and 77 for 0 (Hameed 43*, Burns 31*) need 291 more runs to beat India 191 and 466 (Rohit 127, Pujara 61, Thakur 60, Pant 50, Woakes 3-83)
Shardul Thakur and Rishabh Pant's half-centuries frustrated England and left them needing to complete their highest-ever run chase to take a 2-1 lead in the series, but an unbroken 77-run opening stand between Rory Burns and Haseeb Hameed kept them in with a chance of pulling it off.
India resumed three wickets down with a lead of 171 on the fourth morning, and pressed on towards a significant third-innings total thanks primarily to Thakur and Pant's 100-run stand for the seventh wicket. They lost wickets in clusters but England's bowlers struggled to create regular chances on a flat Oval pitch and India's total of 466 was their highest in the second innings of a Test since 2009.
England's openers were tasked with seeing out 32 overs in the evening session and managed to do so with few concerns. Virat Kohli quickly turned to Ravindra Jadeja, who bowled 13 overs on the trot as he searched for the rough created by the seamers' footholes, but he struggled for consistency in his length. Burns was gritty and defensive but Hameed found some fluency, clipping boundaries off his pads and driving forcefully through the covers to leave England eyeing an improbable win.
Kohli and Jadeja - retaining his new role at No. 5 - resumed after seeing off the second new ball on the third evening and saw off James Anderson and Ollie Robinson's first spells with ease, Kohli pounding Anderson for four through the covers and punching Robinson through mid-off with a strong bottom hand to move into the 40s.
But Chris Woakes' first spell of the day brought two breakthroughs. His second ball was angled across Jadeja and defended into the off side, but both England and umpire Alex Wharf recognised that it had hit the knee roll of his pad first. Jadeja reviewed the decision, but the on-field call was upheld.
Three balls later, the out-of-form Ajinkya Rahane was pinned on the pad while shouldering arms to a nip-backer, but successfully reviewed Wharf's decision, with the DRS showing that it would have bounced over the top of the stumps.
Rahane had no such luck in Woakes' following over: he was set up perfectly, with Woakes gradually pushing his release point wider until Rahane played around another nip-backer that would have crashed into the top of his leg stump. His duck meant his Test average dipped below 40 for the first time in six years and his place will be under severe pressure in Manchester.
Kohli struggled to score as fluently in the second hour of the morning session, tied down by Woakes in particularly, and fell six runs short of his half-century as another England bowling change worked perfectly: Moeen Ali's sixth ball of the day drifted away, spun in and took the outside edge to give Craig Overton a straightforward catch at slip.
That left India leading by 211 runs with four second-innings wickets in hand and with their last two recognised batters at the crease in Pant and Thakur. Both men would have been tempted to play in their usual counter-attacking style but instead batted through to lunch with composure, as England burned their final review on a hopeful appeal for a bat-pad catch.
England toiled after lunch with little success. Moeen struggled for consistency, dropping short with regularity, and while Pant stayed within his shell, Thakur was more than happy to wait for the bad ball and pick off runs with controlled aggression.
Anderson and Robinson were both asked to bowl more than 30 overs in the innings and Thakur treated them with disdain, a straight six off Robinson's offcutter the pick of his shots before he brought up his second half-century of the match. There were occasional chances, with Moeen making a mess of a run-out opportunity and Pant surviving a tight lbw shout when attempting to reverse-sweep Anderson, but they brought up a 100-run stand after taking the lead beyond 300.
They fell within six balls of each other. Thakur was suckered into a drive outside off when Root brought himself on, and after bringing up a measured half-century, Pant skipped down the pitch to try and thump Moeen back over his head but only managed to offer a return catch. His reaction was one of self-flagellation, despondently dragging himself off after smacking his pad with his bat.
Umesh Yadav and Jasprit Bumrah had some fun either side of tea while keeping England's bowlers out in the dirt, hitting 25 and 24 respectively in quick time, but both were caught in the infield by Moeen as Woakes and Overton boosted their figures with late wickets. It left England with two hours to bat in the sunshine and facing a record chase.
But aside from one hopeful lbw appeal from Mohammed Siraj against Hameed, which was turned down both on the field and by the DRS, they struggled to create chances in the evening session. With four members of support staff self-isolating and two senior players in Rohit Sharma and Cheteshwar Pujara off the field nursing niggles, Kohli will need to step up as captain on the final day.
England vs India
ICC World Test Championship
India tour of England
Matt Roller is an assistant editor at ESPNcricinfo. @mroller98