Hazlewood reached 200 wickets in test matches by taking sensational bowling figures of 5-8 and Cummins took 4-21 as India lost eight wickets for only 27 runs inside the first session on the third day.
No. 11 batsman Muhammad Shami injured his right forearm while facing Cummins and was retired hurt as India, which had a 53-run first- innings lead, capitulated on a hard and bouncy wicket against Hazlewood and Cummins.
India's previous lowest ever total in a test innings was 42 made against England at Lord's in 1974. To add to India's worry for the remaining three test matches on the tour, its captain Virat Kohli will not be available. Kohli will be leaving his team to be with his wife, Anushka Sharma, who is expecting his first child.
Joe Burns made an unbeaten 51 off 61 balls and guided Australia to 93-2 after he added 70 runs with makeshift opener Matthew Wade. The left-handed Wade scored 33 before getting run-out in a bizarre fashion when he moved out too far off his crease against Ravichandran Ashwin and wicketkeeper Wriddhiman Saha hit the the ball at the stumps blindly.
Burns, who was struck on the forearm by Bumrah's short ball early in the Australia run-chase, sealed the victory in style with a six over fine leg off Umesh Yadav. Burns also hit seven fours as Australia wrapped up the game in the middle session of the third day.
"It's pretty easy to get motivated against such a top team like India," Burns said.
Kohli said his team collapsed.
"It's very hard to put those feelings to words, we had a lead of 60-odd," Kohli said. "When you work hard for two days and put yourself in a strong position and then an hour puts you in a position where it's literally impossible to win. To be honest there were some good balls but the ball didn't do anything drastic."
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India took a first-innings lead in a test match dominated by pace bowlers after it bowled out Australia for 191 in the first innings. But the extra bounce in the wicket, extracted by Hazlewood and Cummins on Saturday, undid the visitors as none of its batsmen could reach double figures.
Cummins began the Indian slide when he had overnight batsman Jasprit Bumrah caught off his own bowling in the first over of the day and followed it up when Cheteshwar Pujara got a thick edge behind the wicket.
Hazlewood made it 15-5 with his double-wicket maiden first over when he found edges of top-scorer Mayank Agarwal (9) and Ajinkya Rahane's bats and within no time India lost four wickets without a run.
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"Everything went to plan,"" Hazlewood said. When asked if he'd ever had a more perfect day of bowling, he replied: "I don't think so."
Kohli tried to break the shackles with a boundary off Cummins on the off side, before he was caught off the second attempt by Cameron Green off the next ball.
Hazlewood bowled another double-wicket maiden and was on a hat-trick when he removed Wriddhiman Saha and Ravichandran Ashwin off successive deliveries before Umesh Yadav averted the hat-trick ball.
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Hanuma Vehari guided India passed the ignominy of New Zealand's lowest ever score of 26 runs in a test innings with a boundary off Cummins before Hazlewood completed his eighth five-wicket haul in test matches by having the batsman caught behind.