After two days of fluctuating fortunes at the end of a tight series which is level at 1-1, India moved ahead in the morning session when they fashioned a valuable 32-run lead before being dismissed for 332 on the stroke of lunch.
On the liveliest of the four pitches used in the series, Australia's top order struggled against the bouncing ball before Jadeja and Ravichandran Ashwin shared six wickets between them to bowl out the tourists for 137.
Chasing 106 to complete a 2-1 series victory, India were 19 for no loss at the close at the Himachal Pradesh Cricket Association Stadium which is hosting its first test.
Openers Lokesh Rahul (13) and Murali Vijay (six) will resume on Tuesday, with the hosts needing 87 runs to complete a comprehensive win inside four days.
"We claimed three early wickets, which was a bonus for us," Jadeja told reporters.
"We had already taken an upper hand with the 32-run lead. From there, the momentum was with us," added the all-rounder.
Jadeja made 63 and featured in a 96-run partnership with wicketkeeper Wriddhiman Saha after India resumed 52 runs adrift and with four wickets in hand.
Jadeja was hit on the helmet after deflecting a ball from Pat Cummins, who peppered the batsman with short deliveries. The 28-year-old connected with one to smash his fourth six before dragging a delivery from the same bowler on to his stumps.
Steve O'Keefe sent back Kumar in the next over and Cummins (3-94) got rid of Saha, who made 31, with another short delivery.
Off-spinner Nathan Lyon (5-92) dismissed debutant Kuldeep Yadav for seven to wrap up the innings and complete his second five-wicket haul of the series.
Australia's top order struggled from the start.
Karun Nair spilled David Warner at slip off Bhuvneshwar Kumar for the second time in the match but the opener, who has managed just one fifty in the series, was unable to capitalise and was caught behind for six off Umesh Yadav.
Much depended on captain Steve Smith and the highest scorer of the series hit three delectable boundaries before guiding a Bhuvneshwar Kumar delivery on to his stumps to depart for 17.
Umesh (3-29) had Matt Renshaw caught behind as Australia lost three wickets before they had wiped out India's first- innings lead.
Jadeja dismissed Shaun Marsh, Cummins and O'Keefe before Ashwin joined the party, claiming 3-29.
Glenn Maxwell top-scored with a fluent 45 before falling lbw to Ashwin.
"You can hear a pin drop in the dressing-room," batting coach Graeme Hick said.
"Very disappointing day. There's always a chance until the final ball is bowled. But it was hard day today. The boys will be pretty down," the former England batsman added.
(Editing by Ed Osmond)
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