A knock of 30 runs from just 14 balls courtesy of Chris Gayle helped the Windies set New Zealand a victory target of 140.
In reply, after Rob Nicol was dismissed early for 3, Brendan McCullum came to the crease to steady his nation's innings.
However, when he was bowled by Samuel Badree for 22, the Kiwis began to wobble as a couple of quick wickets fell.
Ross Taylor came in to keep New Zealand in contention, but his knock of 61 was only enough to force a Super Over.
The loser of this match will be eliminated, while the winner has a chance of making the semi-finals.
