Chasing a total of 257/9, the Blackcaps lost wickets at regular intervals, but an 80-run stand for the fifth wicket between Neesham and Neil Broom, who scored 48, helped them get past the finishing line with four wickets and 15 balls to spare.
While skipper Latham scored 54, Neesham chipped in with 52 from 48 balls. Mustafizur Rahman was the pick of the Bangladesh bowlers, finishing with 2/33 from 9 overs.
Earlier, Bangladesh rode on the fifties from Soumya Sarkar (61), Mushfiqur Rahim (55) and Mahmudullah (51) to post 257/9 in their 50 overs.
Hamish Bennett picked three wickets for the Kiwis from his 10 overs, conceding 31 runs.
Jimmy Neesham, who also grabbed two Bangladesh wickets, received the Man of the Match award.
Hosts Ireland will face Bangladesh in the next match of the tournament on Friday.