Tommy Smith scored the winning goal with a header at the end of second-half injury time.
"We got what we needed, and whilst at the end of the day it wasn't the best of performances, it gives us a chance," said New Zealand head coach Ricki Herbert. "I'm delighted, they (his players) stuck to their guns and they got something from it."
Chris Killen gave the home side a 10th minute lead, before Cesar Lolohea equalized 11 minutes after the interval.
New Zealand, which went undefeated at the 2010 FIFA World Cup finals in South Africa with draws, will close out Oceania qualifying play with a game against the Solomon Islands on March 26.