Two second half tries by the East African side saw them overhaul the Blitzboks in the final match of the tournament, played at the Harare Sports Club and hard as they tried, mistakes thwarted their comeback efforts as the clock ran out for the South Africans.
They failed to automatically qualify for the Games by finishing outside of the top four at the end of the 2023 HSBC World Rugby Sevens Series earlier in the year.
However, if given clearance from SASCOC, the Blitzboks may get one final opportunity next year in June as 12 teams will fight it out for one spot in the repechage tournament for the Olympics, which will be held in Monaco.
After their clean sweep on Saturday to top Pool A, the South Africans managed to outplay Nigeria (27-0) and Uganda (26-14) in the quarter- and semi-finals respectively, but they ran out of steam and time in the final against Kenya.
Against Nigeria, two tries apiece by Ronald Brown and Shilton van Wyk, and a try by Impi Visser eased them home, but the two tries conceded against Uganda caused some uncomfortable moments early in the second half. With scores tied, a quick brace by Rosko Specman sealed the win.
Kenya scored first in the final, following some sloppy defence by the Blitzboks, but Selvyn Davids struck back from the restart to dot down in the corner.
Christie Grobbelaar then scored following a long-range run, with Ricardo Duarttee's conversion handing the Blitzboks a 12-7 lead.
The second half was tense, but two tries in the final two minutes not only handed the Kenyans the lead, but also the win and a plane ticket to Paris.
South Africa 27 (17), Nigeria 0 (0)
Tries: Ronald Brown (2), Shilton van Wyk (2), Impi Visser. Conversion: Brown.
South Africa 26 (14), Uganda 14 (7)
Tries: Zain Davids, Justin Geduld, Rosko Specman (2). Conversions: Ronald Brown, Selvyn Davids (2)
South Africa 12 (12), Kenya 17 (7)
Tries: Selvyn Davids, Christie Grobbelaar. Conversion: Ricardo Duarttee.