Kuldeep claimed three wickets and Jadeja two as Australia collapsed from 122-1 at lunch to 198-6 shortly after tea on Saturday in a pattern all too familiar for an Australian batting order yet to score a century this series.
At stumps, Peter Handscomb was 28 not out and Pat Cummins was on 25, with Australia still needing another 287 runs to avoid the prospect of the follow-on after India posted 622-7 declared in its first innings.
Jadeja (2-62) struck twice quickly after lunch as India took control.
The spinner first took the wicket of Marcus Harris straight after lunch for 79, playing on to his stumps. It was Harris' second half-century and also the highest score by an Australian this series, underlining the gulf in class between the hosts and India which has scored five hundreds.
Shaun Marsh, whose last century was against England here last year, hit two boundaries before he became Jadeja's second victim, edging to Ajinkya Rahane at slip for eight in the 48th over.
New No. 3 batsman Marnus Labuschagne (38) was then caught spectacularly by Rahane off Shami (1-54) three over later to reduce Australia to 152-4.
On the cusp of the tea break, Kuldeep had Travis Head caught and bowled for 20. He then bowled captain Tim Paine for five in the first over after tea.
Earlier Saturday, Australia progressed untroubled to 72 without loss from its overnight score of 24-0 before an errant shot by Usman Khawaja gave India its first breakthrough in the 22nd over.
Khawaja, who was dropped without scoring by Rishabh Pant late on Friday, wasn't as lucky off Kuldeep (3-71) when on 27 he played a poor shot and chipped a ball to Cheteshwar Pujara at midwicket.
Harris, who scored 70 in Perth's second test, brought up his 50 off 67 balls and looked set to be Australia's first centurion this summer confidently reaching 77 by lunch, but could only add two more runs after the break before he fell to Jadeja to start the host's collapse.
India has all but confirmed its first series victory in Australia by batting for nearly two full days in raising its 622-7 declared with Pujara (193) and Pant (159 not out) scoring centuries.
India leads the four-test series 2-1 after wins in the first test at Adelaide by 31 runs and last week's Boxing Day test to retain the Border-Gavaskar trophy.
Australia won the second test in Perth by 146 runs.