"It turns out that one of the most brilliant pieces of action came from one of the most unlikely guys," commentator Ian Smith said. "When it left the bat I thought six - all over red rover."
Vettori, who retired from international cricket after New Zealand's defeat to Australia in the final, showed little emotion after the catch even though teammates mobbed him.
"Whatever I've done in my 18 years of cricket, that's the only thing I'll be remembered for," he said. "I guess that just encapsulated the whole World Cup. You just never know."
New Zealand bowled the West Indies out for 250 in just 30.3 overs to reach the semifinals.
They went on to face South Africa at Eden Park, and Grant Elliott's six from the penultimate ball to win was every Kiwi's World Cup highlight.
His match-winning 84 not out saw the Black Caps home in dramatic fashion but Elliott benefitted from some good fortune to chase down South Africa's revised target of a 298 in a rain affected match.
On the final ball of the penultimate over, with the Black Caps 284-6, Elliott skied one into the Auckland night and two fielders near deep square leg had eyes on it.
Farhat Behardien ran in from the boundary while JP Duminy came flying in from fine leg, which put Behardien off. He dropped the chance and the pair collided. Meanwhile, Elliott scampered through for two crucial runs.
"It was an easy catch if it wasn't a World Cup semifinal but it's a World Cup semifinal," Ross Taylor said. "Who wants it? I don't think any of them wanted it."
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Communication on the field was extremely difficult given the electric, often deafening atmosphere at Eden Park.
"I wasn't listening to the crowd much. I could hear when catches went up," said Elliott.
And when they went down.
