Apisai Koroisau kicked penalties either side of half time at Wellington Regional Stadium to give Fiji victory in a dogfight of a test match and a ticket to a semi-final against reigning champions Australia in Brisbane next week.
Tonga, who beat Lebanon in Saturday's other quarter-final, will take on England or Papua New Guinea, in the other semi in Auckland.
New Zealand had been looking to get back on track after a stunning upset at the hands of the Tongans last week, the first by a tier two nation against tier one opposition at a World Cup.
One of only three countries to have won the World Cup, the 2008 champions instead found themselves under huge pressure from the off and but for Fiji's decision to tap penalties rather than take the points, could have faced a heavy deficit at the break.
As it was, the two points from hooker Koroisau's successful 15th minute effort were all that divided the teams.
New Zealand halfback Shaun Johnson levelled up the scores from the kicking tee five minutes after the break but the hosts were unable to build on that as winger Jordan Rapana was sent to the sin-bin for a professional foul two minutes later.
Koroisau restored Fiji's lead with his second penalty just after the hour mark and the islanders held out for the remaining 19 minutes to reach the last four of the World Cup for the third straight tournament.
(Reporting by Nick Mulvenney in Sydney; Editing by John O'Brien)
