At the close in Sydney, the visitors were 141-6 with Ravindra Jadeja on eight and Washington Sundar six after they dismissed Australia for 181 in reply to their first innings 185.
Pant signalled his intent by plundering a six on his first ball, and reached 50 after just 29 deliveries with another big shot that cleared the ropes.
Only his 28-ball half-century against Sri Lanka in 2022 was faster for India.
He was finally dismissed for 61 off 33 by Pat Cummins, but Scott Boland was India's main tormenter, taking 4-42.
Pant's exploits set up an exciting finale, with doubts over whether skipper Jasprit Bumrah will take any further part.
He left the field after bowling one over following lunch and left the Sydney Cricket Ground, likely heading for hospital scans. Bumrah was seen returning some 80 minutes before stumps.
In his absence, Prasidh Krishna (3-42), Mohammed Siraj (3-51) and Nitish Kumar Reddy (2-32) stepped up to fill the void and dismiss Australia after they resumed on 9-1.
The hosts will regain the Border-Gavaskar trophy for the first time since 2014-15 if they win or draw.
Yashasvi Jaiswal got India's second innings off with a bang, crunching four boundaries from the opening over by Mitchell Starc.
But the opening partnership was quickly shattered when KL Rahul was bowled for 13 by Boland, whose line and length has been impeccable all series.
He pounced again in his next over with a ball that jagged back to deliver the same punishment to Jaiswal (22).
Virat Kohli desperately needed a big score -- for himself and the team -- but failed to deliver, perishing in the slips for six with his nemesis Boland again the destroyer.
It was likely the 36-year-old's last Test innings in Australia, with India usually only visiting every four years.
Debutant Beau Webster, who top-scored for Australia with 57, claimed Shubman Gill (13) as his first Test wicket, but at the other end Pant was on a mission until an outside edge off Cummins carried to wicketkeeper Alex Carey.
Boland claimed his fourth to remove Nitish Kumar Reedy cheaply.
Dramatic
Australia lost Sam Konstas (23), Marnus Labuschagne (two), Travis Head (four) and Steve Smith (33) before lunch.
In front of his home crowd, Smith looked set to become only the 15th batsman ever and fourth Australian to reach 10,000 Test runs.
But he was made to wait, five runs short, after Prasidh enticed an edge to Rahul in the slips.
The home side resumed after a dramatic final-ball wicket on Friday when Bumrah -- captaining the side after Rohit Sharma was "rested" -- had Usman Khawaja caught in the slips.
That Australian body blow came two balls after Bumrah and Konstas had a tense exchange.
The 19-year-old Konstas padded up again on seven with new partner Labuschagne, who didn't last with Bumrah getting the faintest outside edge to Pant.
A fearless Konstas slammed Bumrah to the boundary, then produced an audacious reverse ramp shot for another four.
But the teenager overplayed his hand and fell to a Siraj outswinger, attempting a drive that flew to Jaiswal at gully.
Siraj removed Travis Head in the same over with another gem to leave Australia tottering at 39-4.
Smith and Webster, in the team for the dropped Mitchell Marsh, began rebuilding the innings before Prasidh dismissed the veteran just before lunch and bowled Carey soon after their return.
India then took 3-4, with Reddy at the forefront, as the tail collapsed.
