It was an agonising result for the Silver Ferns, who will play Uganda for third place on Monday morning while unbeaten Australia and England will contest the final.
Coach Noeline Taurua handed first caps in the white-hot atmosphere to goal attack Amorangi Malesala and centre Tayla Earle.
Both entered the game in the second quarter, replacing the Heffernan sisters - goal attack Georgia and centre Kate.
New Zealand trailed 15-14 at that point before going to the main break 31-30 ahead.
They trailed 46-43 after three quarters and slipped further back before their late charge, inspired by the shooting under pressure of Nweke and two intercepts from Karin Burger, who had shifted back to goal defence from wing defence.
Nweke continued her accurate form by landing 47 of her 50 shots while Malesala played a fine support role on debut, shooting six from seven.
“There were moments, especially at the start and during the third and fourth quarters, when we were down by six but there was a lot of grit shown out there,” Taurua said.
“Our ability to finalise some of those plays at the end, whether it’s a pass through court or the connections are the things we have to learn on the job.
“When it came down to it, we should have got that last goal in. But you’ve got to learn to play to win and we’re not quite there, so all credit to the Roses.”
New Zealand were whistled for 58 penalties to England’s 32, with goal keep Kelly Jury in particular struggling with interpretations before she was replaced ahead of the final quarter.
Midcourter Maddy Gordon wasn’t considered after suffering a head knock at training during the week.
Nat Metcalf was typically influential in the England midcourt while Eleanor Cardwell (35 from 40) and Helen Housby (23 from 26) were reliable in the shooting circle.
Earlier, Australia beat Uganda 63-49 to consign the African side to their third loss.
New Zealand’s sole win at the tournament came over Uganda, 65-60, while they lost their opener to Australia 63-50.
