South Africa, who lost to Kenya in the pool stage, claimed their first victory on the circuit since winning in Port Elizabeth 12 months ago.
France took the bronze medal when they defeated Kenya 28-26 in the third-place playoff.
South Africa fell behind in the final, as they did in each of their three knockout games on Sunday, when German Schulz scored for Argentina, but Speckman and Senatla showed explosive speed to guide the hosts to a 17-7 lead.
The South Americans fought their way back into the contest on the stroke of halftime when Axel Muller burst through for their second try.
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Yet the home side, roared on by a crowd of 52,600 at the Cape Town Stadium, controlled territory in the second period and Ryno Benjamin made the game safe late on after Senatla had scored his second try.
South Africa had beaten France 21-12 in the semi-finals, while Argentina edged Kenya 24-22.
There are 10 rounds in the World Rugby Sevens series before the sport makes it first appearance at the Olympics in Rio de Janeiro next year.
The next tournament on the circuit will be in Wellington, New Zealand on Jan. 30-31.
(Reporting by Nick Said; editing by Toby Davis)
