Bashir, who was in South Africa for an African Union summit, was allowed to go even though a Pretoria court had issued an order banning him from leaving until the end of a hearing on whether he should be detained under a global arrest warrant.
The court later said he should have been arrested to face genocide charges at the International Criminal Court (ICC).
"We've studied ... the final judgement that was presented last week and we have decided that government is going to appeal," government spokeswoman Phumla Williams said.
"We will be submitting our appeal affidavit within the prescribed 14 days and that's where all the reasons will come out."