Ten dropped catches cost New Zealand their first two games against the West Indies and Sri Lanka, and they faced elimination with two group games remaining if they hadn't beaten Ireland, who are also winless.
The Kiwis recovered from 3-10 to post 6-140 on a last-ball six by Suzie Bates, the storied allrounder making her first appearance in the tournament only after fellow retiree-to-be Sophie Devine became ill.
New Zealand pacer Bree Illing bowled out with a sterling 1-18 but Ireland captain Gaby Lewis and Orla Prendergast played around her and Jess Kerr (0-23) in a partnership of 110 into the 18th over.
They put Ireland in sight of an historic first World Cup win after 19 defeats, but Prendergast was caught near the boundary on 59. Lewis was still there on 58, though, until the next and 19th over when she gave Bates a catch at cover.
That left Ireland needing 15 runs from the last over to win and Bates conceded only 10 and restricted the Irish to 4-136.
"It has been a disappointing start to the tournament so today was just about body language and turning up," said New Zealand captain Melie Kerr, the player of the match. "We need to turn and believe we can still win games of cricket."
Ireland chose to bowl first and Prendergast and Aimee Maguire reduced New Zealand to 10-3 in the fourth over.
Melie Kerr picked up the pace with 30 and Brooke Halliday and Izzy Sharp combined for 62 in 50 balls.
Bates came in at No.7 for the first time in her T20 career and contributed with a cameo 19 off 12 balls, including a last-ball six that cleared long leg that ultimately proved decisive.
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