The hosts were desperately clinging on at 227-8 by tea on day four with Alex Carey 30 not out, battling for survival with an improbable 534 needed for victory.
Mohammed Siraj has 3-51 and stand-in skipper Jasprit Bumrah 3-42 on a wearing Perth Stadium pitch.
The visitors declared at 487-6 late on day three after Yashasvi Jaiswal's superb 161 and an unbeaten 100 from superstar Virat Kohli, then rocked Australia with three wickets.
Rookie Nathan McSweeney was out for a duck, nightwatchman Pat Cummins fell for two and an out-of-touch Marnus Labuschagne departed for three.
Faced with a mountain to climb, Australia resumed at 12-3 with Usman Khawaja on three and Steve Smith yet to score.
Khawaja added just one when he mistimed a pullshot off Siraj and got a big top edge.
It was caught by backtracking wicketkeeper Rishabh Pant, who became the most expensive player in Indian Premier League history Sunday at the Twenty20 tournament's lucrative auction.
Head survived a loud lbw shout on seven, but a review found it was missing leg stump and he battled on, smacking seven boundaries for his 17th Test half-century.
At the other end, Smith was felled by a Harshit Rana bouncer that slammed into his midriff, needing a spell lying on the ground to recover.
He was able to get up and play on, and like Head came through an lbw review on 12.
The 62-run partnership was ended by Siraj, with Pant taking another neat catch after Smith edged a delivery he had to play on 17.
Head teamed up with Marsh in an 82-run stand to temporarily raise hopes before Bumrah again worked his magic.
Seemingly destined for a century, the largely untroubled Head feathered to Pant with Bumrah's loud double fist-pump showing how ecstatic he was at the breakthrough.
All-rounder Marsh kept the scoreboard ticking over with Carey but fell after dragging a wide Nitish Kumar Reddy delivery onto his stumps.
Mitchell Starc, top-scorer in the first innings, was out 12 on the cusp of tea to the spin of Washington Sundar.
