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Chahar smashed Kiwi pacer Adam Milne for a 95-metre six in the final over of the innings.
The Kiwis had reached Jaipur for the series opener less than 24 hours after losing the World Cup final to Australia in Dubai.
Axar Patel claimed 3/9 off his three overs.
McClenaghan's response to an Indian fan went viral on Twitter.
Harshal won the player-of-the-match award on his international debut against the Kiwis.
Openers K L Rahul (65) and captain Rohit Sharma (55) added 117 off just 80 balls to set up the Indian win.
Kohli has relinquished his T20 captaincy and from Wednesday's opening T20 International against New Zealand, Rohit will be the full-time skipper in the shortest format going into next year's World Cup.
Tim Southee will captain the T20I side.
New Zealand may have lost the final of the T20 World Cup but they won the hearts of fans world over by the decency and decorum with which they carried themselves on and off the field.
Opening batter David Warner scored 53 off 38 balls while Mitchell Marsh remained unbeaten on 77 from 50 deliveries to see Australia through and spark wild celebrations among his teammates.
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