Fiji went on to beat close rivals New Zealand 35-19 to lift the Cup at the USA Sevens for the first time since 2007 at round five of the World Series.
South Africa bounced back to beat tournament hosts USA 31-0 in the playoff for third place. Frankie Horne, Kwagga Smith, Cecil Afrika (2) and Rayno Benjamin were the scorers.
The South Africans now have 93 points after two Cup wins and successive third place finishes so far this season. They lead New Zealand by five points with Fiji third on 86. The top four teams at the end of the World Series qualify for the 2016 Olympic Games in Rio.
"We were our own worst enemy against Fiji in the first half, but after halftime we showed how we can play. When things don't go your way in the semis I think it is important to come out and still give it your best so you can collect the maximum on offer in the third place playoff, which we did. I think we did very well against a fired up USA to win third place," explains Brown.
South Africa now return home to prepare for the next World Series event next month in Hong Kong.
The Springbok Sevens skipper added: "We will do a lot of introspection but we will not be drastic and change a lot of things. We have a talented squad and we played some good rugby this weekend, but only in short bursts. Hong Kong is a fantastic venue and a great tournament and we will work very hard during training so we are fully prepared for that challenge."
The Springbok Sevens recovered from an unforgettable start to come within one score of clinching a dramatic win against Fiji in the first of the two semi-finals. Semi Kunatani (2), Viliame Mata and Tuwai Vunisa scored for the islanders before halftime.
Horne was yellow carded for a deliberate knock down while skipper Kyle Brown was sent to the sin bin for an early tackle. Afrika, Seabelo Senatla and Philip Snyman all scored as the Blitzboks turned up the heat in the second half, but they left their rally just a little too late.
New Zealand defeated the USA 26-12 in the other semi-final to set up Cup final clash with rivals Fiji.
Round 6 of the World Series takes place from 27 to 29 March in Hong Kong.
The HSBC Sevens World Series standings after Round 5 in Las Vegas:
1. South Africa 93 points
2. New Zealand 88
3. Fiji 86
4. Australia 71
5. England 68