Batting first, the Black Caps may have felt they left a few runs out on Saxton Oval, after a late-innings collapse.
Devon Conway top-scored with a fluent 56 off 34 balls, before he was run out by a brilliant throw from the deep.
Mitchell blasted 41 from 24 deliveries.
But New Zealand lost wickets cheaply late to end on 177 for nine, with Sharmar Springer bowling two tidy overs at the death.
Then, after a third consecutive tail-end batting revival in the series, the West Indies fell just short of the 178 target in a dramatic final over.
A late partnership between West Indies tailenders Springer and Romario Shepherd — who came together at 88 for eight, with their side looking dead and buried — was not quite enough.
Nos 10 and 8 respectively in the batting order, the pair put on a record 9th wicket partnership of 78 off just 39 balls to bring the equation down to just 13 off 7.
But Jacob Duffy held a sharp caught and bowled off the last ball of the 19th over to get rid of Springer for 39, and Kyle Jamieson then dismissed Shepherd for 49 runs in the final over, as, looking to find a couple of boundaries to win the game, the Guyanan skied the ball high to Mitchell on the long off boundary.
Game 4 in the series is again in Nelson today, starting at 1.15pm.— RNZ / additional reporting Allied Media
