Chapman reached his fourth ODI half-century from 58 balls and went on to score 64 in a 75-run fifth-wicket partnership with Michael Bracewell as New Zealand chased down the West Indies' 161 with 19.3 overs remaining.
New Zealand held a 2-0 lead after winning the first match by seven runs and the second by five wickets.
The West Indies chose to bat first after winning the toss but managed only 161 on a pitch at Seddon Park which had more menace than its relatively placid appearance suggested.
First through Matt Henry with the new ball, then with effective use of the short ball and later with Mitch Santner's spin, New Zealand dismissed the tourists in 36.2 overs.
"It was quite a slow surface so probably looked a bit different to how you want to go about your work here at Seddon Park," Henry said. "It's a small ground all round so you want to be taking wickets throughout to put teams under pressure.
"Thankfully, we could keep doing that with some bowling from the whole team."
New Zealand faced real difficulties in their chase, stumbling to 3-32 after Will Young was brilliantly caught by Khary Pierre in the 11th over in one of the highlights of the match. Pierre dived full length at short cover to pick off Young's drive off Matthew Forde in mid air.
New Zealand were 4-70 when Tom Latham was out for 10 after grafting for 25 balls. The West Indies kept New Zealand under early pressure with tight lines and hustle in the field.
But Chapman and Bracewell eased the pressure in a partnership which was the largest of the match. Chapman reached his 50 with a six in an over in which he took 17 runs from the bowling of Forde. It was his fifth score of 50 or more in his last seven innings. Bracewell finished 40 not out from 31 balls.
West Indies captain Shai Hope was happy to bat first on what looked to be a docile pitch at Seddon Park, dry and sparsely grassed. But there was a little swing early and, when that abated, New Zealand's persistence with a shorter length paid off as batters made a start but often fell to soft dismissals.
Henry dismissed Ackeem Auguste and Keacy Carty within the space of three balls in the fifth over after the West Indies had made a bright start. The tourists scored at a run a ball through the first 10-over power play but lost three wickets in that period which exposed the middle order.
Four wickets fell for 36 runs in the next 10 overs — the West Indies were 7-96 — and the tail again was called on to come to the rescue.
Roston Chase bolstered the innings with 38 from 51 balls before falling to a short ball from Henry which flew to Bracewell at extra cover.
Khary Pierre, playing for the first time in the series, showed some defiance with an unbeaten 22, including two sixes before Henry finished off the innings.
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