Needing 18 to win in the Super Over, West Indies rode on Chris Gayle and Marlon's Samuels pyrotechnics to finish the game with a ball to spare at the Pallekele International Cricket Stadium which saw a tied game with both teams finishing on 139 in regulation time.
While Gayle started the chase in the Super Over with a huge six over long-off off a no ball by Tim Southee, Sameuls completed the job with a maximum over deep mid-wicket.
With three loses from as many games, New Zealand have crashed out of the tournament. On the other hand, West Indies' fortune will now hinge on the result of the other Group 1 match between hosts Sri Lanka and England.
The game went into the Super Over after New Zealand made heavy weather of a modest 140-run chase.
Ross Taylor, who gave New Zealand hope with his aggressive batting in the Super Over, played a captain's knock but his unbeaten 62 went in vain as chasing 140, New Zealand managed 139 for seven in their alloted 20 overs.
Sunil Narine shone with the ball for West Indies with figures of three for 20 from his four overs, which included the penultimate over of the New Zealand innings in which the off-spinner conceded just three runs besides picking up a wicket.
Earlier, pacers Doug Bracewell and Tim Southee came up with superb bowling efforts to help New Zealand bundle out West Indies for a lowly 139.
New Zealand's decision to field first produced the desired result as the Kiwi bowlers managed to frustrate the opposition batsmen throughout. Southee (3/21) and Bracewell (3/31) claimed three wickets each, while Nathan MuCullum took two for 19 to end Windies' innings in 19.3 overs.