The Tigers were 166 for four after 33 overs chasing 271 runs to win when this report was filed, needing 105 runs in the last 17 overs.
Latham was dropped off the third ball of the innings, before he had even opened what was eventually a costly account, when the returning Nasir Hossain dropped an utterly simple sitter at square leg off Mashrafe's bowling.
Mustafizur Rahman (1-46) provided the breakthrough for Bangladesh with the wicket of Luke Ronchi (2) in the fourth over but Latham and Neil Broom (63) ensured New Zealand ended the first powerplay at 60 for one. They were 156 for the loss of one wicket at one stage before Bangladesh pulled things back.
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Everton have never won at the Emirates Stadium while their last win at the old Highbury was way back in 1996. Aaron Ramsey is fit despite being forced off against Sunderland in midweek with a thigh strain.
It began with a home whitewash of Zimbabwe before they showed significant signs of improvement by reaching for the first time the quarterfinals of the World Cup held the following year in Australia in New Zealand. Nasir then got a taste of the medicine he made his skipper sample when Soumya dropped Broom, on 56, at extra cover.
The win in Dublin, with 10 balls to spare, was Bangladesh's first one-day worldwide defeat of New Zealand away from home soil.
Captain Latham top-scored for the third successive Black Caps worldwide, posting a patient 84 off 92 balls after his team were asked to bat.
That is what transpired, thanks mainly to the death bowling of Rubel Hossain, Mustafizur and Shakib and a tightening up in the fielding.
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Eric Bailly will miss the final after he was sent off in the semi-final second leg against Celta Vigo at the Theatre of Dreams. However, Ajax triumphed 2-1 at Old Trafford in the most recent meeting between the sides in the 2011-12 Europa League last 32.
New Zealand will want to stay unbeaten in the series keeping with their top-dog status among the three teams in the mix. They were the unsafe trio of Jimmy Neesham, Mitchell Santner and Colin Munro.
Corey Anderson (24) struggled for timing and none of the remaining batsmen managed double figures as the innings conjured just 103 runs from the last 20 overs.
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Such operations need to be within 12 nautical miles of an island or territory to constitute a challenge under global law.