After captain Jasprit Bumrah won the toss, India lost six wickets in the middle session, including flamboyant Rishabh Pant (37) and impressive debutant Nitish Kumar Reddy (41).
Hazlewood took 4-29 while Mitchell Starc, Mitchell Marsh and Pat Cummins all claimed two wickets each.
After a crushing 3-0 home series defeat by New Zealand, India sprung a surprise at the toss by dropping veteran spinner Ravichandran Ashwin, allrounder Ravindra Jadeja and middle-order batsman Sarfaraz Khan.
Coupled with the absence of opener and regular skipper Rohit Sharma following the birth of a child, and without injured number three Shubman Gill, it left India with a fragile batting lineup which Australia exploited.
After four wickets fell before lunch, including Virat Kohli for five, Pant and Dhruv Jurel needed to hang around.
But it was not to be, with Jurel, preferred to Khan, surviving barely 10 minutes before succumbing to Australian T20 skipper Marsh on 11.
Filling the all-rounder role left vacant by the injured Cameron Green, Marsh tempted Jurel into a thick edge that carried to third slip Marnus Labuschagne.
He struck again soon after to remove Washington Sundar, gloving to wicketkeeper Alex Carey to leave the visitors staring down the barrel on 73-6.
Pant led a mini recovery, steering them past 100 after surviving a difficult dropped catch.
Partnered by Reddy, the runs began flowing more freely including the first six of the match.
Cummins finally ended Pant's exploits, taken sharply at slip by Steve Smith before Hazlewood removed Harshit Rana (7) and Bumrah (8).
Earlier, Yashasvi Jaiswal departed without scoring in the third over, edging an attempted drive off Starc to debutant Nathan McSweeney, who did well to collect low at gully.
With Gill missing, left-hander Devdutt Padikkal came in at three and survived a big lbw shout almost immediately.
Australia's pace attack was at full throttle and it proved slow going for India.
The nervy Padikkal faced 23 deliveries without scoring before his luck ran out when Hazlewood steamed in and enticed an edge that was easily taken by Carey.
That brought Kohli to the crease, in dire need of a big score after only two Test centuries in the last five years and with questions mounting over whether he still warranted selection.
Despite a stellar record in Australia he was never comfortable and lasted just 12 balls before fending off a climbing Hazlewood thunderbolt on five that took an edge to Usman Khawaja at slip.
India's woes worsened when opener KL Rahul feathered to Carey on 26.
The visitors have won their last two Border-Gavaskar trophy series in Australia.
Perth is the first of five Tests in the series.
