Jadeja took 5-65 while fellow slow bowler Washington Sundar had 4-81 as the Kiwis were spun out soon after tea on Friday.
Daryl Mitchell top-scored with 82 and Will Young made 71 after New Zealand won the toss.
Akash Deep trapped in-form opener Devon Conway lbw for four before captain Tom Latham added 44 runs with Young for the second wicket.
Gaining sharp purchase against the left-handers, Sundar bowled Latham for 28 and Rachin Ravindra for five.
In hot and humid conditions, Young was caught at slip off Jadeja in the 45th over. Three balls later, Jadeja bowled Tom Blundell for a duck, before also bowling Glenn Phillips for 17 just before tea.
After the interval, Jadeja trapped Ish Sodhi lbw for seven and bowled Matt Henry for a two-ball duck. Sundar then had Mitchell caught at first slip.
Sundar finished things off by removing Ajaz Patel lbw for seven, completing a collapse of 7-76,
New Zealand won the first two tests in Bengaluru and Pune by eight wickets and 113 runs respectively, registering a landmark first Test-series win in India.
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