The trio of Eddie Dawkins, Sam Webster and Ethan Mitchell were beaten by Great Britain by just over a 10th of a second in the final.
New Zealand finished with a time of 42.542 seconds with Great Britain ahead of them finishing with Olympic record time of 42.440 seconds.
New Zealand had easily overcome Germany in their heat in a new Olympic record time of 42.545 seconds.
The medal is New Zealand's third today, after the rowing men's pair of Hamish Bond and Eric Murray defended their Olympic title to claim the country's first gold of the Games and kayaker Luuka Jones won silver.
Meanwhile the New Zealand men's and women's team pursuit have both advanced to the next round.
The men were fourth fastest in qualifying with the women finishing fifth fastest.