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, but were eventually rolled for 291 after two gripping sessions.

Virat Kohli admitted it was hard to stay calm as the hosts threatened to achieve something that history suggested was impossible, with the highest chase in an Adelaide Test remaining Australia's 6-315 in 1902.

Shaun Marsh's 60, Tim Paine's 41 and a wagging tail that showed oodles of conviction and composure ultimately failed to prevent Australia suffering their first Test loss to India at home since 2008.

Monday's 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 with just a single ball remaining before tea.

Kolhi celebrated passionately and screamed in delight.

Nathan Lyon dropped to his haunches and cut a distraught figure on 38 not out, having been dropped on seven.

Paine, who suffered a finger injury while batting that he insists is not serious, admitted the defeat was "pretty shattering" but backed his team to bounce back in the second Test that starts in Perth on Friday.

"I don't think many people thought we'd get as close as we did today and certainly didn't think we'd win," Paine told reporters.

"It showed our group that anything is possible if you are prepared to stick it out ... we take a lot of confidence from today. The way our guys fought was sensational.

"It's going to be a really tight series.

"If you want to be a good team, you have got to be hard to beat and today we were.

"We made India work really hard, I think we made them work harder than they thought they were going to."

Kohli disagreed, saying India knew Australia "were not going to throw in the towel".

"I wouldn't say I was cool as ice, but you just try not to show it," he said.

"They gave it a go, tried their best but we executed our plans - eventually."

Kohli was clearly frustrated by a third-umpire verdict that overturned Cummins' dismissal on zero plus some dropped catches and an Ishant Sharma no-ball that all had the potential to prove costly.

Kohli snaffled an edge at first slip to remove Cummins for 28 and reduce Australia to 9-259.

The captain celebrated Cummins' dismissal by furiously 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.

Cheteshwar Pujara, who scored a total of 194 runs from 11 hours at the crease, was named man of the match.

Australian Associated Press