In pursuit of an uphill victory target of 405, England finished the day at 87 for 2, thanks to a dogged defiance from skipper Cook and young Haseeb Hameed, who brought back memories of Hashim Amla and AB de Villiers' 'batathon' at the Feroz Shah Kotla, last year.
It was fascinating final two sessions where Cook (54, 188 balls) and 19-year-old Hameed (25, 144 balls) added 75 for the opening stand but more importantly consumed 50 out of the 150 overs that England require to bat in their second innings.
The captain's resistance finally ended when Ravindra Jadeja trapped the skipper leg before with the final delivery of the day. England require 318 runs more on the final but they would be very happy if they could even manage a draw.
India have three full sessions tomorrow to grab the remaining eight wickets and take a 1-0 lead in the five-match series.
More From This Section
If Cook's experience came to the fore, Hameed defied his age and showed temperament of highest order batting for more than three hours.
In desperation for something special, Ashwin finally delivered a shooter right in middle and leg in his 13th over of the day, and there was no way that Hameed could keep it away as he was trapped right in front for an agonising end.
The opening partnership also provided a study in contrast to their struggling Indian counterparts as the strategy was just to block and delay the inevitable.
England reached 50 in 37.1 overs, a tad faster than their slowest of 38.5 overs in the second innings against Sri Lanka at The Oval in 1998.
(REOPENS DEL 28)
On the docile surface, Jadeja, however, followed up his five wickets in the first innings for a match haul of nine wickets.
But Handscomb and Marsh, whose 53 came off 197 balls, ensured that the Australians stayed afloat to fight another day.
Earlier in the morning session, India looked good for a win by claiming Renshaw and first innings unbeaten centurion Smith in the space of four balls.
But thereafter Australia got a grip of the situation while India struggled to break the partnership on an eased-out track of the JSCA International Stadium.
But just when the crowd went silent, Ranchi's icon Dhoni lifted the atmosphere when he made an entry and watched the match from the second session onwards.
Back from Delhi after leading Jharkhand in the national one-dayers, Dhoni was seen in a relaxed mood and waved at the cheering crowd.
On the field, Smith's ploy to give his top four the maximum workload when they bowled 206 overs meant that Indian could not score quick runs.
But Jadeja gave the side a good shot at a result by using the wearing track brilliantly. He cleaned up Smith after Ishant's dismissal of Renshaw in an eventful 29th over.
The breakthrough came after Smith and Renshaw held on for one and half hours in the morning.
Smith, who scored a spectacular 178 not out in the first innings to lead Australia to 451, misjudged a probing line from Jadeja with the ball pitching between middle and leg.
Just in the previous over, Sharma set it up brilliantly to dismiss the left-handed Aussie opener.
Noticing some activity near the sight-screen, Renshaw decided to pull out. Ishant, having already bowled four overs at a stretch, threw the ball in his follow through out of frustration.