Kohli struck a 104-ball 105, his second successive and fourth overall ODI century, to guide India to 276 before an abrupt collapse in 49 overs.
Part-timer Yuvraj Singh (three for 43), his lanky spin colleague Ravichandran Ashwin (three for 50) and paceman Shanthakumaran Sreesanth (three for 30) then shared nine wickets as New Zealand folded for 236 in 45.2 overs.
Ross Taylor (66) tried his best and down the order, Nathan McCullum (35) and Kyle Mills (32) threw their bats at everything but that was not enough for the tourists, beaten 1-0 in the recent test series, to avert defeat.
It was the 22-year-old Kohli who set the tone for India, walking in after openers Murali Vijay and Gautam Gambhir had put on 44 and guiding the team to the 250-mark before he fell in the 44th over.
For New Zealand, Andy McKay (four for 62) and Kyle Mills (three for 42) scythed through the lower half of the Indian batting order with the last five wickets making just 26.
Jaipur hosts the second match on Wednesday.
