Nchout turned inside Ria Percival in the 95th minute and superbly curled in the winner to put Cameroon through.
They were heading out after Aurelle Awona sliced a volley into her own net with 10 minutes remaining for 1-1.
Nchout had put her side ahead in the second half with a lovely, dinked goal.
Although Cameroon finished on three points, the same as Nigeria from Group A, they went through courtesy of having scored just a single goal more (3) than their African counterparts.
The Netherlands finished top of Group E with a 100% record after a 2-1 victory over Canada, who finish in second place.
It was a rather uneventful game up until the 57th minute when Nchout brought the game to life by turning inside of two New Zealand defenders and keeping her cool with a smart finish.
Cameroon were the better side thereafter but the game suddenly turned when Awona inadvertently converted into her own net, at which point both were staring at a heartbreaking exit.
New Zealand, who remain without a World Cup win at the 15th attempt, could have made it through to the knockout stages but Betsy Hassett's low header was brilliantly tipped round the post by goalkeeper Annette Ngo Ndom.
Olivia Chance screwed wide from inside the box on 87 minutes before the excellent Nchout made space in the box and slotted in with the last kick of the game deep in injury time to spark wild celebrations in Montpellier.
Goal-shy New Zealand - the stats
- Cameroon progress to the last 16 for the second successive tournament.
- Cameroon's second goal came with 94 minutes and 49 seconds on the clock - the latest winning goal (excluding extra time) in World Cup history.
- Cameroon are now six World Cup games without a clean sheet, conceding nine goals.
- New Zealand are still without a win in their 15 World Cup matches - an ongoing record.
- New Zealand have only scored eight goals in their 15 matches in the tournament.
- Aurelle Awona scored the sixth own goal of the tournament - equalling the record for one tournament set in 2015.