After two days of fluctuating fortunes in the finale of an evenly-matched series tied at 1-1, India surged 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, Australian top order struggled against the bouncing ball before Jadeja (3-29) 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 stumps 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 looking set to complete a comprehensive win inside four days after thoroughly dominating the third day.
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Earlier, 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 Cummins, who peppered the batsman with short deliveries. The 28-year-old connected on one of them to slog 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 India innings and complete his second five-wicket haul of the series.
Australia's top order let them down when they walked out to bat.
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.
Much depended on skipper Steve Smith and the highest scorer of the series hit three delectable boundaries before guiding a Bhuvneshwar Kumar delivery onto his stumps to depart for 17.
Umesh (3-29) then had Matt Renshaw caught behind as Australia lost three wickets before they had wiped out India's first innings lead.
Jadeja dismissed Shaun Marsh, Pat Cummins and Steve O'Keefe before Ashwin joined the party, claiming 3-29.
Glenn Maxwell topscored with a fluent 45 before paying the price for offering his pad, and no shot, to Ashwin.
