Earlier, they lost 28-0 to Australia in the quarter-final after finishing second in their pool on Saturday.
Their inability to break down opposition defences and too many missed tackles told the story of the Blitzboks' weekend, as they are still without triumph in th Mother City since 2015.
In the first match of the day, Australia pinned the Blitzboks in their own half and starved them of possession, as the South Africans' charge to replicate their Dubai triumph from a week ago came unstuck.
The Blitzboks put in a great defensive effort in the first half, especially when playing with six men when Quewin Nortje was yellow carded, and they managed to keep the Aussies out.
However, that changed late in the first half when Australia scored two converted tries to go into the break 14-0 up, as the South Africans hardly had any meaningful possession in the half.
It was 21-0 less than a minute into the second half when the Blitzboks turned over possession from a promising counter-attack, leaving them with a mountain to climb, in five minutes.
They could not do what was necessary and crashed out of the Cup competition in disappointing fashion, with Australia scoring a fourth try to rub salt into the wounds of the home team.
More salt came their way against New Zealand, who scored twice to take a 10-0 lead into the break.
A Ronald Brown try not long after the start of the second half, converted by Justin Geduld, gave the home crowd something to cheer about, but New Zealand then stepped on the gas and scored three further converted tries for an emphatic victory.
Scorers:
Springbok Sevens 0 Australia 28 (14)
Australia - Tries: Hayden Sargeant, Nathan Lawson (2), Dietrich Roache. Conversion: Maurice Longbottom (3), Roache.
Springbok Sevens 7 (0) New Zealand 31 (10)
Springbok Sevens - Try: Ronald Brown. Conversion: Justin Geduld.
New Zealand - Tries: Ngarohi McGarvey-Black, Cody Vai (2), Sione Molia, Akuila Rokolisoa. Conversions: Tepaea Cook Savage (2), Rokolisoa.