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India, hunting their maiden test series win in Australia after 11 fruitless tours spread across almost 71 years, hold a 1-0 lead in the four-match showdown after nervously turning the screws on Monday.
Australia resumed at 4-104 on day five, requiring a further 219 runs to overhaul an imposing target of 323 and complete a record-breaking chase.
Virat Kohli, clearly frustrated by a third-umpire verdict and some dropped catches that had the potential to prove costly, snaffled an edge at first slip to reduce Australia to 9-259.
Kohli celebrated Pat Cummins' dismissal by hurling the ball at the ground, rightfully bullish given the victory equation was as simple as restricting the final-wicket stand to 62 runs or less.
The post-lunch session was extended by half an hour as India searched for their last wicket, finally delivered when offspinner Ravichandran Ashwin had Josh Hazlewood out edging in the final over before tea.
As Kohli celebrated passionately and screamed in delight, Nathan Lyon cut a distraught figure on 38 not out.
Expectations of a dramatic finish grew as Australia fought hard in two gripping sessions, showing conviction that would please coach Justin Langer but ultimately paying the price for lapses in recent days.
Ashwin fluffed his lines before striking late, Paine suffered a finger injury that would have major ramifications if serious, and chirpy keeper Rishabh Pant made it an Indian record 11 dismissals for the match while missing two crucial chances.
The visitors created a breakthrough whenever Australia looked to have shifted momentum in their pursuit of a result that history suggested was impossible, with the highest chase in an Adelaide test remaining the 6-315 that Australia achieved in 1902.