The duo featured in a marathon 365-run stand, India's highest fourth wicket partnership against any team, applying themselves on a pitch where the occasional ball kept low.
Martin Guptill (17) and Tom Latham (six) managed to see through the final nine overs to push New Zealand to 28 for no loss at stumps, still 529 runs behind on a fast-deteriorating track.
Rahane had not looked comfortable against the rising delivery on Saturday and the right-hander was predictably subjected to a barrage of short-pitched balls by Matt Henry after India resumed on 267-3.
One such delivery had the batsman in two minds whether to duck under a bouncer which smacked him in the helmet.
Rahane took a single off Trent Boult to bring up his century and looked more confident against the spinners, stepping out to hit three sixes, including two off Jeetan Patel's bowling.
Kohli took a single off Henry to bring up his second test double hundred before being hugged by partner Rahane, while rival skipper Kane Williamson came forward to shake his hand.
After enduring three sessions without success, New Zealand finally got a breakthrough when Patel struck in the first over after tea, trapping a tired Kohli lbw.
Kohli's career-best 211 included 20 boundaries, of which there were a couple of crisp down-the-ground drives.
Rahane had already bettered his previous high score of 147 and was looking good for his maiden double century when he edged Boult and BJ Watling took a diving catch behind the stumps to end his 381-ball stand.
Rahane's stellar knock included 18 boundaries and four sixes in the first test match to be staged at Indore's Holkar Stadium.
Rohit Sharma (51 not out) helped himself to his third fifty of the series before Kohli declared.
(Reporting by Amlan Chakraborty in New Delhi; Editing by John O'Brien and Toby Davis)
