"We thank SuperSport United for their support towards the goals of the Association and its national teams".
Carlos Queiroz has a long-term contract with the Iranian national team.
Baxter will transition into the role once he completes his current duties as coach of SuperSport United after the SAFA reached an agreement with the South African club to acquire the services of the former Finland coach.
South Africa are now second in Caf's World Cup qualifying Group D - level on points with Burkina Faso - while they have been pooled alongside Nigeria, Libya and the Seychelles in the race to reach Cameroon in two years' time.
There have been plenty of delays, with SuperSport's Absa Premiership chase suffering as a result because - by Baxter's own admission - the players were affected by his imminent departure.
In a break with an SAFA tradition of making major announcements at media conferences, confirmation that Baxter had been appointed came via an emailed statement.
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"The club has given them time until today to finalise the issue".
SAFA and Supersport United have agreed to work together to ensure that the Baxter gets time off post the final league match on 27 May to prepare Bafana Bafana for the upcoming AFCON match against Nigeria.
Finally - it was a long wait, almost five months, to be precise.
South Africa's next competitive fixture is on the weekend of June 9-11, when they start their 2019 African Nations Cup qualification campaign away in Nigeria.
However, up to this stage there is still no sign of a new face at the helm with Soccer World Cup qualifiers taking place early next month.
Matthews has on several occasions expressed his dissatisfaction at how the delay has affected his team and planning for next season.