The South Africans - who should slip down the overall HSBC World Rugby Sevens Series standings - won their match against Hong Kong China 31-12, but paid a heavy price in that fixture, which was sandwiched between the two Antipodean clashes, as three players picked up injuries that ruled them out of the match against New Zealand.
The Blitzboks will play Ireland at 05h56 (SA time) on Sunday in one of the ninth-place quarter-finals.
The opening match against Australia - who were two log points behind the Blitzboks at the start of the tournament - knocked their aspirations of keeping their opponents below them on the standings.
A strong start did not yield points for the Blitzboks despite good ball possession and playing in the Aussie half of the field.
Instead, it was the men in yellow and green who scored three minutes in when Henry Paterson ran in from wide out. It was a killer blow as up that point, Australia hardly landed a shot, but held a 5-0 lead at the break.
To add to the Blitzboks' woes, Ricardo Duarttee was yellow carded early in the second half and from the resulting play, Nathan Lawson scored. Dietrich Roache added a third to ease the Wallabies into their first win of the day.
The Blitzboks needed a big score to bounce back against Hong Kong China, but conceding two second half tries in their 31-12 victory did not help their quest to build a healthy points difference.
Impi Visser scored first before Mfundo Ndhlovu got a brace, with the second coming from a great run in which he stepped five defenders. All three tries were converted by Duarttee and the 21-0 lead at the break looked promising.
An early second half try by Max Denmark cut into the SA lead, before tries by Sebastian Jobb and Ethan James pushed them further in front again, but a late try by Liam Herbert undid some of that good work. Yellow cards to Duarttee and Visser further compounded the SA pressures.
The final match against New Zealand was stacked with obstacles.
The Blitzboks not only had to beat New Zealand by more than 30 points in order to get into the Cup quarter-finals, but also had to do so without Shaun Williams, Tiaan Pretorius and Duarttee, who all picked up injuries in the second match.
They deserve credit though for giving it a proper shot.
In an entertaining match that went from one side of the field to other, the Blitzboks took the lead five minutes in when Ndhlovu dotted down. The centre stepped inside his defender following patience on attack and when the NZ defence dropped off the pace, he could pounce to score under the poles. Jaiden Baron converted and the Blitzboks were 7-0 up.
From the restart New Zealand replied, attacking from their own line and finally worked Dylan Collier over for a converted try and a 7-7 scoreline at the break. The second half was as tough, but when the men in black scored with two minutes left - following a slick lineout move - the writing was on the wall.
South Africa 7 (7), New Zealand 12 (7)
South Africa - Try: Mfundo Ndhlovu. Conversion: Jaiden Baron.
South Africa 31 (21), Hong Kong China 12 (0)
South Africa - Tries: Impi Visser, Mfundo Ndhlovu (2), Sebastiaan Jobb, Ethan James. Conversions: Ricardo Duarttee (3).
Australia 19 (5), South Africa 0