The hosts, who elected to bat first, rode on an unbeaten 167-run partnership between Kohli (103) and Ajinkya Rahane(79) to gain control of the innings after early losses.
Kohli recorded his 13th Test ton in 48 matches after a string of low scores in his last seven innings since his first Test double hundred against West Indies in July.
Kohli eyes whitewash, improvement in personal form
The top-order batsman, who tackled the New Zealand bowlers with excellent footwork, hit the ball to all parts of the Holkar Stadium.
Rahane, who seemed to gain confidence from the free-flowing Kohli at the other end, brought up his fifty -- the tenth of his Test career -- with a six off Jetan Patel.
The 27-year-old Kohli worked hard in the initial part of his innings, but grew in stature after getting past fifty and his assured stroke-making soon dominated the New Zealand bowling.
Kohli tapped a rising delivery from Matt Henry to run a quick single that saw a direct hit at the non-strikers end, but the star batsman scampered home and celebrated his hundred.
India's Kumar out of third Test against New Zealand due to back strain
New Zealand skipper Kane Williamson, who returned to lead the side after missing the second Test with a viral infection, juggled his bowling options but Kohli and Rahane held fort.
The hosts did not lose a single wicket in the space of 34 overs in the final session as the batting duo entertained the raucous home crowd.
India lost both its openers in the morning session with returning Gautam Gambhir (29), who was playing Test cricket a two-year gap, letting go of a good start.
Pace spearhead Trent Boult cut short Gambhir's breezy knock, which was studded with three fours and two sixes, after trapping the left-handed batsman lbw on an incoming delivery.
Boult, Santner and Patel, who struck first after sending back Murali Vijay for 10, shared a wicket apiece in the 90 overs of play.
The hosts, who took a 2-0 lead after winning the second Test in Kolkata on Monday, have set their sights on a whitewash against the tourists.
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