England have been set 378 to beat India and level the delayed Test series 2-2, which would eclipse their current record run-chase of 359 - the Ben Stokes-inspired triumph at Headingley.
An intrepid century stand between openers Alex Lees (57) and Zak Crawley (46) carved a huge chunk of the target, before Root (76 not out) and Bairstow (72no) continued their magnificent form to leave the hosts 3-259 at the end of day four.
With 119 needed and belief implausibly high after knocking off 277, 299 and 296 against New Zealand in the last month, England go into the final day as favourites.
A full house of more than 23,000 will be in attendance to watch the latest unlikely victory charge of the Stokes-Brendon McCullum era after Warwickshire followed the example of Nottinghamshire and Yorkshire by offering free entry for the final day.
Root and Bairstow came together with 109 on the board and with India swarming after taking out the top three for two runs either side of tea.
India captain Jasprit Bumrah had ripped out Crawley and Ollie Pope to get his side's adrenaline pumping, before Lees was run out following a miscommunication with Root.
While England's fearless approach to batting continues to astound, they would have been staring at an even bigger challenge had it not been for an excellent effort with ball in hand. India held all the cards on 3-125 at the start of play, but lost their remaining seven wickets for 120.
Cheteshwar Pujara carved Stuart Broad to slip for 66 and first-innings centurion Rishabh Pant made a mess of reverse sweeping Jack Leach on 57 as both failed to turn good starts into match-winning scores. Stokes also did a fine job of rounding up the lower order with a barrage of short balls, taking the last three to claim four for 33 - his best figures in almost five years.
Lees then charged his third delivery of the innings and swatted Mohammed Shami for four through mid-wicket to set the tone of England's reply. He and Crawley put on 100 before the end of the 20th over - England's fastest century stand for the first wicket.
India pegged England back with that trio of wickets but Root and Bairstow rebuilt before taking the attack back to India.
England joy was tempered by an allegation of racial abuse from the stands which prompted a investigation by ground officials. The England and Wales Cricket Board expressed concern.
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