First Test, Adelaide Oval (day five):
Australia 235 & 291: Marsh 60; Shami 3-65, Bumrah 3-68. Ashwin 3-92
India 250 & 307: Pujara 71, Rahane 70, Lyon 6-122
India won by 31 runs
India clinched a dramatic 31-run victory in the first Test against Australia to claim only their sixth Test win down under.
The home side, chasing 323 for victory, resumed day five in Adelaide on 104-4.
They fell to 187-7 after Shaun Marsh (60) and Tim Paine (41) were dismissed but with Nathan Lyon making 38, the lower order gave Australia hope.
They had reached 291 when last man Josh Hazlewood was caught off spinner Ravichandran Ashwin.
It is India's first victory in a Test series-opener in Australia and their first Test win in the country since 2008.
The next match of the four-Test series will start in Perth on Friday at 02:20 GMT.
"Obviously it's a great feeling, we've never taken a lead in a series in Australia," said India captain Virat Kohli.
"I think our fast bowlers came in and bowled with great heart but we knew (Australia) were not going to throw in the towel."
'It's pretty shattering'
Australia are playing their first home Test series since March's ball-tampering scandal and were without the suspended Steve Smith, David Warner and Cameron Bancroft.
They had bowled India out for 250 in the first innings but conceded a 15-run first-innings lead to the tourists after being dismissed for 235.
India made 307 in their second innings before Jasprit Bumrah, Mohammed Shami and Ashwin got three wickets apiece to help bowl their side to victory.
"It's pretty shattering," said Australia captain Tim Paine.
"Really proud of the fight of the lower order but I think over the five days we had a number of opportunities to get ahead of the game but failed to capitalise on any of them and India thoroughly deserved to win."
The tourists' wicketkeeper Rishabh Pant took 11 catches in the match to equal the record held by England's Jack Russell and South Africa's AB De Villiers.
He could have had the record to himself but dropped Nathan Lyon off Bumrah.