The visitors, led by Kusal Perera's 102 and Niroshan Dickwella's inventive and entertaining 76 were dismissed for 326 in 49 overs, chasing New Zealand's imposing 371 for seven, which was anchored by Martin Guptill's 138 and Kane Williamson's 76.
Neesham, who is battling with three other players for an all-rounder's spot for the May 30-July 14 World Cup in England and Wales, however, overshadowed Guptill's 14th ODI century with a six-hitting assault at Bay Oval in Mt. Maunganui.
The 28-year-old, who had not played for New Zealand since June 2017, was on three at the start of the 49th over bowled by Thisara Perera and proceeded to smash five sixes.
He then finished New Zealand's innings with a sixth six off Nuwan Pradeep, having scored 47 from 13 balls.
Neesham then dismissed both of Sri Lanka's openers after Dickwella and Danushka Gunathilaka (43) had given their side a superb start with a 119-run stand in less than 18 overs.
He also snapped up the dangerous Dinesh Chandimal for his third wicket to reduce Sri Lanka to 211 for four in the 33rd over just as they were preparing themselves for an assault.
Perera, however, kept the visitors in the chase with his fourth one-day century, before he fell in the 46th over with his side needing 74 runs to win the game.
The second game of the three-match series is at the same venue on Saturday.
(Reporting by Greg Stutchbury; Editing by Sudipto Ganguly)
