Bafana Bafana went into the game as group leaders but a 1-0 defeat at the Cape Coast Sports stadium saw Ghana leapfrog them into first position and advance to the play-offs, which will be held in March next year.
Ghana's winning goal came from a 33rd minute penalty, awarded after Daniel Amartey was fouled in the box, allowing captain Andre Ayew to cooly convert for the only goal of the game.
South Africa FA president is Tebogo Motlanthe is adamant the game was manipulated by the referees and has called for a replay.
"The match officials have decided the game, which is not what is supposed to happen," Motlanthe said in a statement posted on the SAFA's official Twitter page on Monday.
"We will be writing to both CAF and FIFA – firstly to investigate how the game was handled and secondly to challenge some of these decisions.
"We have already registered with the match commissioner that we will be making a formal complaint."
"We are very disheartened by their actions and we can't let it destroy the players – when we have the chance to challenge, we will challenge."
"We feel very much robbed because it is not an isolated incident of a penalty.
"There are just a lot of questionable decisions by match officials and we will get an expert to dissect other incidents, so we can make a strong case."
In the qualifiers for the 2018 FIFA World Cup in Russia, South Africa had one game against Senegal replayed after FIFA established that the referee was guilty of manipulating the game.
Motlanthe added: "Of course we are looking at the precedent whereby FIFA ordered us to replay Senegal and we think that if justice needs to be served the same decision should be taken against this Ghana game."
