Ollie Pope made a magnificent century, his best innings for England, to keep the tourists alive after three days of the first Test against India in Hyderabad.
Pope's epic 148 not out dragged England to 316-6 in their second innings, 126 ahead and with hope of pulling off a famous win.
The vice-captain received brilliant support from Ben Foakes, who made 34 in a sixth-wicket stand of 112 that first ensured England were not beaten with two days to spare, then built a handy lead.
Though Foakes was bowled by Axar Patel, Rehan Ahmed survived to the close on 16 not out. Perhaps crucially, Pope was dropped on 110 and will return on Sunday with the opportunity to set India a testing target.
All of this after Jasprit Bumrah's scintillating spell of reverse-swing bowling helped reduce England to 163-5 and in danger of wasting a good start to the day.
India managed only 15 runs in the morning session, bowled out for 436, with Joe Root taking two wickets in two balls to end with 4-79. Overall, the hosts lost their last three wickets for no runs but still had a lead of 190.
If England are to achieve something remarkable, they may have to hope for an improvement in the condition of leading spinner Jack Leach, who had his left leg heavily strapped and was able to bowl only one over.
England dream of the unthinkable
England could still lose this Test by a wide margin, but this was their best day of the contest, much improved from the slog of day two and certainly giving some encouragement for the rest of the series.
The tourists did well to limit India from their overnight 421-7. Root got Ravindra Jadeja and Bumrah in successive deliveries, while Ahmed bowled Axar for 44 in the next over.
But it is with the bat where England showed real grit, with all of the top three going some way to answering the questions that hang over them in this part of the world.
Ben Duckett survived Ravichandran Ashwin for the first time in his Test career on the way to 47 and Zak Crawley, with such a poor record against spin in Asia, made an attractive 31.
They were outshone by Pope, who made one of the great hundreds by a batter visiting this country, especially given the quality of the attack, the conditions, match situation and the fact this is his first Test since June following shoulder surgery.
It evoked memories of Alastair Cook's century in a losing effort in the opening Test of England's tour here in 2012, one that started the shift in momentum to set them on course for a series win.
It is still a huge ask for England to get anything from this Test. The pitch seems to have lost pace and bite, so defending anything other than a sizeable target will be difficult for England's inexperienced attack against India's powerful batting.
But England have a chance, which is more than the had at the beginning of the day.
Pope gives England a prayer
Pope's effort is all the more remarkable after his frenetic one from 11 balls on the opening day, when he looked every inch like a man who had not batted competitively since the second Ashes Test in June.
Even early in this innings the England vice-captain was far from his fluent best, often swiping, mis-timing and missing altogether.
But he came through the Bumrah spell and watched wickets fall around him, clipping, flicking and driving his way past 50 for the first time in 10 innings in this country.
When Ben Stokes was out before tea and with England still 27 behind, there was every chance the tourists would be beaten before the day was out.
Foakes, also unconvincing in the first innings, was a composed ally, defending with a straight bat and scoring with the occasional sweep.
Pope grew in stature, reaching his fifth Test century by working Jadeja through the leg side, then regularly reverse-scooping the spinners over slip.
Pope was reverse-sweeping when Axar made a meal of a catchable chance at point and although Axar got one to scuttle through Foakes, all of Ahmed, Tom Hartley, Mark Wood and Leach are capable of helping Pope push England further ahead on Sunday.
Brilliant Bumrah blows England open
England had reached 89-1 at lunch, making assured progress on a diet made up almost exclusively of spin. But fast bowler Bumrah, on straight after the break, blew the game open by getting a ball only 16 overs old to reverse-swing, with devastating results.
Bumrah, who is remarkably playing only his fifth Test on home soil, should have had Duckett lbw, only for India not to review. In the next over, he got through Duckett's flashing drive with a hooping in-swinger and, in the the next, pinned an unbalanced Root leg before.
It was riveting stuff, the sort of Test cricket that makes every delivery an event, creating an electrifying atmosphere for the 28,750 spectators inside the Rajiv Gandhi Stadium.
Jonny Bairstow did well to repel Bumrah, but lost his off stump playing no shot to a straight one from Jadeja, who had got the ball before to turn sharply. Stokes, subdued for six from 33 balls, played all around an Ashwin half-volley that turned to hit off stump.
India were surging, only to be repelled by Pope and Foakes. As the spinners struggled to make an impression, Bumrah was summoned for two ominous spells in the evening session.
England resisted and now face the challenge of the second new ball, available after three overs on the fourth morning.
'England back in this'
Ex-India wicketkeeper Deep Dasgupta: "That was a great day of Test cricket. England won the first and third sessions, India the second. You can't get a better Test day.
"England are back in the game. Another 40 or 50 runs and we could have an interesting day four or five.
"India are still favourites but England are back in this Test for sure."
England batter Joe Root on TNT Sports: "It was a brilliant day of cricket for us. The way we played and went about it from the very beginning - we were very clinical with the ball and didn't let them capitalise on what they did yesterday.
"The opening stand with the bat was brilliant. It could have been tricky, but they [Crawley and Duckett] have been so crucial for us over the past couple of years. There were partnerships after that too, when we had to get through that tricky phase when it was reversing - some of us didn't quite manage that.
"But the way Ollie played was an absolute masterclass on how to bat in this part of the world. It was truly phenomenal."
