The result means the Blitzboks failed to win any of their pool games, which ruled them out of the running for automatic Olympic Games qualification.
They can still make it to the Paris 2024 showpiece, but will probably have to qualify through Africa.
After going down by 19-15 to Great Britain first up, the USA scored two late tries to draw 26-26 with the Blitzboks, who played much better in their second clash of the day.
The Blitzboks could still make it into the top eight at Twickenham, but they had to beat New Zealand, who had already secured the HSBC World Rugby Sevens Series title for the season.
The Kiwis started well and took a 20-7 lead at the break, courtesy of a hat-trick by Akuila Rokolisoa, with Impi Visser scoring late in the half for South Africa.
New Zealand then added another two tries - both by Xavier Harris - after the break to seal the 32-21 victory, although the Blitzboks refused to give up as Mfundo Ndhlovu and Ryan Oosthuizen added their names to the score sheet.
Against the Eagles, their first two tries came from kicks. First Oosthuizen chipped ahead, James Murphy gathered and put Christie Grobbelaar away as the front row combined in the most unlikely fashion.
Two minutes later, another kick led to Shilton van Wyk's five pointer and it was 19-0 soon thereafter when Dewald Human tapped quickly after a mistake by USA and raced away for the points.
USA came back with converted tries by Kevon Williams and Lucas Lacamp to make it a five-point game at the break.
Ndhlovu capitalised early in the second half from some weak tackling by the USA for South Africa's fourth try to stretch the lead to 26-14, but the Eagles stayed in touch when Williams crossed for his second with three minutes to go.
Pita Vi's converted try then levelled the scores with half a minute to go.
In their pool opener earlier on Saturday, Great Britain cleverly ran down the clock at the end of their clash with the Springbok Sevens to win by 19-15.
Both teams ended the match with three tries apiece, but better goal-kicking from tries scored closer to the posts was the difference in the end.
The Blitzboks will also rue a slow start, which allowed Great Britain to take a 19-5 lead at the break - with Femi Sofolarin grabbing a brace - as the team in red controlled possession and attacked with patience.
Sofolarin's first try was followed almost immediately by a five-pointer from Api Bavadra as the South Africans failed to control the restart, which gave GB a 12-0 lead.
Siviwe Soyizwapi hit back for the Blitzboks from his team's first real opportunity with ball in hand in the first half, which was also his 150th World Series try for South Africa, but Sofolarin had the last say before the break to hand his team a 14-point lead.
The second half was much better for the Blitzboks, with Oosthuizen and Shaun Williams scoring great tries, but in the end Great Britain controlled the ball well after a restart mistake from the Blitzboks to take the win.
South Africa 15 (5) Great Britain 19 (19)
South Africa - Tries: Siviwe Soyizwapi, Ryan Oosthuizen, Shaun Williams.
Great Britain - Tries: Femi Sofolarin (2), Api Bavadra. Conversions: Kaleem Barreto (2).
South Africa 26 (19) USA 26 (14)
South Africa - Tries: Christie Grobbelaar, Shilton van Wyk, Dewald Human, Mfundo Ndhlovu. Conversions: Ricardo Duarttee (3).
USA - Tries: Kevon Williams (2), Lucas Lacamp, Pita Vi. Conversions: Steve Tomasin (3).
South Africa 21 (7) New Zealand 32 (20)
South Africa - Tries: Impi Visser, Mfundo Ndhlovu, Ryan Oosthuizen. Conversions: Ricardo Duarttee, Ronald Brown (2).
New Zealand - Tries: Akuila Rokolisoa (3), Che Clarke, Xavier Harris (2). Conversion: Tepaea Savage.