"People were anxious to get in and just surged ahead. Fortunately it was a contained incident and calm was restored quickly," city spokesman Pieter Cronje said.
All the injured were South African and officals have prevented people from entering the fan zone at Cape Town's historic Grand Parade where Nelson Mandela delivered his first speech as a free man after his release from prison in 1990.
The Grand Parade viewing area is the largest of 10 special fan zones scattered across South Africa's Western Cape province.
Cronje said an estimated 16,500 people had streamed to the venue in Cape Town's city centre to see a concert and fireworks display ahead of Africa's first hosting of the tournament.
Police spokesman Frederick van Wyk said police tried to warn the public through loudhailers that the viewing area was full, but they continued pushing forward.
"People at the back then kept moving forward and a crush ensued," Van Wyk said in a statement. Three people were injured, one man suffered a broken leg and two women sustained broken ankles.
At least 15 people were injured last Sunday when fans tried to force their way into a match between Nigeria and North Korea.
Excitement among South Africans is reaching fever pitch with less than 24 hours to the host nation's opening game and tournament kick-off against Mexico in Johannesburg.
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