Neesham made 107 to support the efforts of first-day centurion Kane Williamson and help the Kiwis make 508-7 before declaring shortly before the close on day two.
Windies opener Chris Gayle, playing his 100th Test, ended unbeaten on eight as he and Kieran Powell survived nine overs to guide the hosts to 19-0, with the latter being dropped by Peter Fulton.
Williamson added only eight to his overnight total before becoming the third New Zealand man to fall, for 113, and Ross Taylor (55) and captain Brendon McCullum (seven) followed in the morning session.
However, Neesham came together with BJ Watling, who made 89, to put on a 201-stand for the sixth wicket, and some lower-order clubbing from Tim Southee took the Black Caps past 500.
Spinners Sulieman Benn and Shane Shillingford toiled away at Sabina Park and eventually finished with three wickets apiece for the cost of a combined 287 runs, on another long day in the field for the Windies.