Chasing a daunting 441, the world's top side crashed to 107 all out early in the final session after left-arm spinner Steve O'Keefe's match haul of 12 wickets, handing number-two ranked Australia the upset win inside three days.
O'Keefe's heroics enabled Australia to register their first Test win on Indian soil since 2004 as they took a 1-0 lead in the four-match series.
The 32-year-old returned with impressive figures of 6-35 as he got good support from off spinner Nathan Lyon, who bagged six wickets at India's newest Test venue.
Lyon got the final wicket of Jayant Yadav (55) as the second-ranked Australian side got into a celebratory huddle.
O'Keefe made India falter twice on a turning track as another 6-35 in the first innings saw the hosts bundle out for 105.
A wily O'Keefe got the big wicket of Indian skipper Virat Kohli as he bowled the star batsman for 13, stunning the home crowd into silence.
It was a rare failure for Kohli, who was out for a duck in the first innings, after scoring double hundreds in four successive series since July.
Earlier on Saturday Australian skipper Steve Smith struck his first century in India before the visitors were bowled out for 285 in an extended morning session.
Smith made 109 to compound India's misery after O'Keefe had put the visitors on top with his heroics on day two at Pune's Maharashtra Cricket Association Stadium.
Mitchell Starc also played a big part after scoring 61 in Australia's first innings of 260 after the visitors won a crucial toss.
Starc then took two wickets in India's first essay, getting Kohli out for nought. Starc also hit a breezy 30 in the second innings.
Young opener Matt Renshaw's gritty 68 on day one also signalled Australia's intent of putting up a show on India's home turf.
The 20-year-old Renshaw fought an upset stomach to score crucial runs in challenging conditions.
The teams now head to Bangalore for the second Test starting March 4. The final two games are scheduled in Ranchi and Dharamsala.
