White Ferns spin bowler Anna Peterson took two wickets and a run out in the final over of the match to give the home side the victory.
After being asked to bat first by the West Indies, the New Zealander's got off to a flier thanks to opening batter Sophie Devine.
Getting off to a lighting quick start, Devine scored 41 off 18 balls before being bowled by West Indian spinner Hayley Matthews.
White Ferns skipper Suzie Bates carried her bat through the innings scoring an unbeaten 52.
Matthews did a great job of choking the New Zealanders in the middle overs to stop any momentum the Kiwi's started off with. She finished with figures of 3-24 off her four overs.
The Kiwi's had an equally impressive start with the ball as the Windies struggled through to 43-3 off seven overs.
Spinners Amelia Kerr and Leigh Kasperek were particularly impressive taking the White Ferns first four wickets as the West Indian innings began to crumble.
Kasperek finished with figures of 3-31 and Kerr with 1-19 off her four overs.
The visitors middle order comeback then began as Deandra Dottin, Merissa Aguilleira, and Kycia Knight all made good contributions.
Aguilleira top scored for the visitors with 38 not out.
Down to the final over with the West Indies needing just five runs, the bowl was given to Peterson. The Kiwi spin bowler took two wickets in the final over and finish with figures of 2-25 off her four overs.
Game four of the series will also be played at Pukekura Park, before the teams move onto Seddon Park in Hamilton for the final Twenty20.
New Zealand 134/7 off 20 overs (Bates 52*, Devine 41, Matthews 3-24 off 4) beat the West Indies (Aguilleira 38*. Knight 28, Kasperek 3-31, Peterson 2-25, Kerr 1-19) by one run.
Full scorecard here: https://scoring.nzc.nz/livescoring/match5010/scorecard.aspx