Chasing 292, after skipper Kane Williamson's fine 148 for New Zealand, West Indies were well poised at 2-130 from 20 overs but collapsed after the dismissals of Shimron Hetmyer (54) and Chris Gayle, who was dropped three times in his innings of 87, which included six sixes. They even took to Twitter to heap praise on the New Zealand side.
Their innings bore striking similarities to New Zealand's for a time as they lost two early wickets, Boult ousting Shai Hope (1) and Nicholas Pooran (1), before a third-wicket partnership put the pressure back on the bowling side.
Kieron Pollard and Dwayne Bravo were among like-for-like replacements for Russell in West Indies' reserves, but they have opted for a possible opener in Ambris, who was the second-highest scorer in the tri-series against Bangladesh and Ireland, with 278 runs in four innings at an average of 92.66 and strike rate of 101.83. With 33 runs needed from the final three overs, he hit three consecutive sixes as part of 25 runs off Matt Henry in the 48th over.
Sheldon Cotrell for West Indies was the pick of the bowlers with his spell of 4-56.
New Zealand's nail-biting five-run victory was set up by two players, and it shows in ESPNcricinfo's Smart Stats numbers. His knock was studded with nine fours and five sixes.
Victory saw New Zealand return to the top of the table and all but secure their place in the semi-finals as a heart-breaking defeat left the West Indies on the verge of elimination.
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"Everyone believed we could get over the line", said Brathwaite. But, an attempted pull saw Trent Boult take a stunning catch just inches away from the ground.
A New Zealand win seemed a long way off when Sheldon Cottrell trapped Martin Guptill lbw with the first ball of the match and followed it up by bowling Colin Munro with a flawless yorker later in the over.
Ex-New Zealand captain Jeremy Coney on BBC Test Match Special: "Kane Williamson is someone who is very hard to keep at the non-striker's end - he seems to have a way of scoring twos so he always gets back on strike, and not at a huge risk either".
Said Neesham: "I knew he didn't get it off the bat".
"I just knew how much everyone in the dressing room wanted it". "All in all I would give West Indies their kudos for playing well, they found themselves in a bad position and then they fought back". "Our record's been really good, and I think the moment the other night was something that probably got to everyone".
West Indies, champions in the first two editions of the World Cup 1975 and 1979, have just one win from six games at this edition and they next face an unbeaten India in Manchester on Thursday.