Both teams tied at the end of the see-saw group one match, New Zealand scored 17 runs off the over sent down by Marlon Samuels.
Samuels returned to score the winning run, hitting a six off the fifth delivery of the erratic Tim Southee over to knock New Zealand out of the tournament.
Earlier put into bat, West Indies were bowled out for 139 in 19.3 overs with Chris Gayle top-scoring for them with a fluent 30 off 14 balls. New Zealand seam bowlers Doug Bracewell (3-31) and Southee (3-21) shared six wickets between them.
Chasing the target, New Zealand struggled against the spin of Sunil Narine (3-20) but skipper Ross Taylor's unbeaten 62 helped them to finish on 139 for seven and force the Super Over.
"When the first ball is a no-ball and goes for a six, it ruins the over a lot," a dejected Taylor said in the post-match presentation ceremony. "That's cricket, we are going home now."
"It was a good game, probably we should have won in normal time. But credit to West Indies, they came back and fought hard. To score 18 off five balls in Super Over is pretty awesome," Taylor added.
West Indies squandered a great start provided by their top order to finish on a disappointing 139 all out after being 86 for three at the half-way mark.
Gayle hit three fours and two sixes while Samuels (24) and Keiron Pollard (28) also chipped in with useful cameos but rest of the batting order could not capitalise on the start.
Bracewell provided New Zealand with the early breakthrough, dismissing Johnson Charles and Andre Russel in his first spell returning to claim the wicket of Pollard to finish with figures of three for 31.
Southee, who had removed Gayle, claimed two more wickets as West Indies managed 53 runs in the second half of the innings while losing seven wickets.
New Zealand's chase never really gained momentum until Taylor came out to bat, hitting three sixes and as many boundaries in his unbeaten 40-ball knock to tie the score and force the Super Over.
Off-spinner Narine put a break on the New Zealand run-rate, capturing three wickets for 20 runs and was assisted by leg-spinner Samuel Badree (one for 18) and seam bowler Ravi Rampaul (one for 23).
"He delivered today. We all know what he is capable of doing, like bowling in pressure situations," West Indies captain Darren Sammy said of Narine.
"And also Marlon Samuels. He's suffering from bad shoulders and that's why he doesn't bowl too many overs."
It was New Zealand's second Super Over loss in the tournament, following Thursday's defeat against Sri Lanka.