The men in black will face Australia in the quarterfinals, at 11.48pm Sunday (NZ time).
Cape Town - Follow the action LIVE from the Cape Town Sevens, the second event on the 2018/19 World Rugby Sevens Series calendar.
The Blitzboks, who finished sixth at the Dubai Sevens last weekend, were grouped in Pool A alongside New Zealand, Samoa and Zimbabwe. Werner Kok scored from the ensuing set piece and Justin Geduld made no mistake with the conversion. New Zealand struck twice before the break to take the lead at 14-7.
Branco du Preez won a breakdown penalty two minutes from time.
If that wasn't enough drama, New Zealand, in the last play of the match went from the halfway line and through multiple phases and scored to seal an incredible win and their path in the Cup quarter final berth.
Earlier, South Africa made a ideal start to their Cape Town 7s campaign with two victories from their opening two matches to instill optimism in a lively Mother City.
The Blitzboks made a big statement against Zimbabwe with a massive 43-0 win to make it two-out-of-two on Day One of the Cape Town Sevens.
The Blitzboks made amends for their defensive failings when they sacked a Kiwi ball-carrier behind the tryline after half-time.
Shujaas were on the back foot from the opening minute with all-time IRB World Sevens series top tries scorer Dan Norton bagging a hat-trick for England in the encounter.
The other quarterfinal matchups will see South Africa take on Scotland, Fiji play Spain and the U.S. clash with England.