The 31-run victory in Adelaide was India's first series-opening win on Australian soil.
Images from Day 5 of the first Test between Australia and India in Adelaide on Monday.
India crushed a lower-order insurrection to secure a nerve-jangling 31-run victory in the first Test against Australia on Monday, raising the South Asian nation's hopes of a breakthrough series win Down Under.
The tourists despatched Australia's last four batsmen in the middle session of the final day at Adelaide Oval to send India's 'Swami Army' of travelling fans into raptures in the sparse fifth day crowd.
Faced with a docile wicket and tenacious tail-end slogging, Virat Kohli's team bowled with patience and determination to claim India's first win in the country since Anil Kumble's men beat a Ricky Ponting-led Australia by 72 runs in Perth in 2008.
It was also India's first win in a Test series-opener in Australia and they will head to Perth for the second of the four scheduled contests full of confidence.
Spinner Ravichandran Ashwin took the final wicket of Josh Hazlewood for 13 to dismiss Australia for 291, with Nathan Lyon unbeaten on 38 after a valiant chase of 323 for victory.
"These things happen in Test cricket... but it's important to stay calm at that point," Kohli said in a post-match interview. "They gave it a go, they tried their best but I think we executed our plans eventually."
After suffering defeat in their first home Test without the suspended Steve Smith and David Warner, Tim Paine's Australia will go into the second match with grave fears about their batting confirmed.
On a pitch that presented few demons, only two home batsmen managed scores above 50 across the match with a majority of victims falling due to poor shot selections.
Kohli's side, however, banked 123 and 71 from man of the match Cheteshwar Pujara, with Ajinkya Rahane chipping in a vital 70 in the second innings against a world class seam attack.
India's seamers were the ones making the difference early on day five, however, removing Australia's last two batsmen in Shaun Marsh (60) and Travis Head (14), leaving wicketkeeper Paine and tail-ender Pat Cummins to scrap desperately to lunch.
Australia resumed after the break 137 runs shy of the victory target, but reduced the margin by only one run before Paine flashed his bat at a short ball to be caught by fellow wicketkeeper Rishabh Pant after skying a top edge.
Mitchell Starc kept the terraces entertained with a typically swashbuckling innings of 28 to push Australia's deficit to below 100 runs.
However, Mohammed Shami broke through to remove his fellow paceman, with wicketkeeper Rishabh Pant catching the edge to equal the world record of 11 dismissals shared by AB de Villiers and Jack Russell.
No team had scored more than 315 runs in a successful run chase at Adelaide Oval but Nathan Lyon and his fellow tail-enders battled hard to break new ground before falling just short.