Despite it being their first loss of the tournament, New Zealand finished third behind Australia and South Africa, who both had superior net run-rates.
The White Ferns got off to a reasonable start thanks to a 43-run opening wicket stand between Suzie Bates (16) and Sophie Devine (40) before Bates fell in the seventh over.
Sara McGlashan and Devine got New Zealand through to 70 for before wickets began to tumble with seven falling in the space of 44 runs. The White Ferns were held to 114 for eight - their lowest total of the tournament.
Marizanne Kapp was the key bowler for South Africa, finishing with figures of three for 23 from four overs.
Mignon de Preez (51) led South Africa in their chase, winning with two balls left in the final over.
