After the visitors had opted to bowl at the toss, left-arm seamer Arshdeep ran through the home team's batting order to record figures of 4-70 as the Black Caps were bowled out for just 185.
It was a sorry tail of woe for the New Zealand's batting unit as opener Tim Seifert top-scored with 38, while it took a late cameo of 34 from number 11 Blair Tickner to lift them past the 180-run mark.
Besides Arshdeep, fellow seamers Gurnoor Brar and Anshul Kamboj also impressed with the ball, taking three and two wickets respectively, as the hosts faced 40.2 overs.
The tourists lost vetaran opener Rohit Sharma for a duck early in the run-chase, before a rain delay meant they had to chase a revised target of 143 in 32 overs.
Kohli put all fuss of the reaching the target to bed with a crisp half-century, and was well-supported by his skipper Shubman Gill (37) before both lost their wickets near the end.
The second ODI of the five-match series will take place in Wellington on Saturday.
New Zealand fast bowler Jacob Duffy believes Virat Kohli remains one of the most difficult batters to contain, after witnessing the Indian star produce a match-winning performance at Eden Park.
New Zealand batter Devon Conway has been ruled out of the first ODI against India after he took a blow to the head after colliding with teammate Matt Henry during training.
New Zealand claimed a morale-boosting seven-wicket victory over India in the fifth and final T20I in Hamilton on Sunday.
India have added seamer Auqib Nabi to their ODI squad for the five-match series in New Zealand following a hamstring injury to Prasidh Krishna.
