The highest successful run chase at the Brisbane venue was in 1951, when Australia finished 236-7.
The other star with the ball for India was Shardul Thakur, who picked a four-for after playing a key hand with the bat as well on Sunday.
Hazlewood was the pick of the Australian bowlers and was rewarded for his tight line with a ninth five-wicket haul in a test innings.
Although most of the Australian batting order got starts, only Steve Smith converted and even he fell for 55 when surprised by a Thakur short ball.
Mitchell Starc takes a catch to dismiss Rohit Sharma on Saturday in Brisbane.
Coach Justin Langer will lament Australia's collapse of 3-4, which included the wickets of well-set batsmen Paine (50) and Cameron Green (47), as well as their inability to post a more imposing total.
India lost wickets at regular intervals and Australian bowlers were relentless from the word go.
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But the hosts continued batting when play resumed in gloomy conditions in the evening session, adding a further 51 runs for three wickets.
Starc's new-ball spell lasted three overs before he was replaced by Pat Cummins, who fired out Shubman Gill with a pinpoint delivery that found the edge and flew to Steve Smith at second slip.
But a 39-run ninth-wicket stand between Lyon and Starc staved that off and helped Australia pass the 350 mark before Sundar struck again close to lunch. Siraj, leading the India attack in just his third test in the injury-enforced absence of Jasprit Bumrah and spinner Ravichandran Ashwin, returned 5-73 for his career-best figures. They were put on the back foot by Smith who got to his 31st half-century in quick time. But replays clearly confirmed the dismissal.
Marnus Labuschagne was once again their hero as he scored a scintillating ton to take Australia to a score of 369 runs in the first innings.
David Warner compiled 48 from 75 deliveries before he was out in the following over, trapped lbw by Sundar to make it 91-2.
Australia need to win to regain the Border-Gavaskar Trophy but, with more rain forecast for Tuesday, India will fancy their chances of surviving for the draw they need to keep the silverware.