The squad was finalised on Monday, two days before they left for Bangladesh, after Arendse initially declined to approve the list which contained only four black players.
"Cricket South Africa, the custodians of the sport in the country, must take full responsibility for not handling the issue in a manner that preserves the integrity and reputation of the sport, the pair said in a statement issued by CSA on Friday.
"In this regard, the President and the CEO of Cricket South Africa wish to express their deep and sincere regret at the manner in which the issue leading up to the selection of the team to tour Bangladesh, has been handled.
"We issue our sincerest apology to the entire nation... for the manner in which the matter had been handled. We accept responsibility for what happened, and wish to commit ourselves to ensuring that it will never happen again."
CSA policy states that six black players should be included in a 14-man squad. The squad was eventually released without Arendse's approval.
Robin Peterson's late inclusion in place of injured fellow spinner Paul Harris took the number of black players in the tour squad to five.
The statement added CSA was still committed to its transformation policy "aimed at ensuring that the imbalances of the past are redressed as soon as possible.
"...We lastly wish to publicly acknowledge that mistakes were made in the process of the selection of the team, and that any finger pointing is not going to assist in restoring any damage or harm that may have been caused to the game of cricket over the recent days.
"Accordingly, the President and the CEO commit to working together within the legal and constitutional structures of Cricket South Africa with a common goal and purpose, with a view towards ensuring the restoration of the good name and reputation of Cricket South Africa, and that the game is administered for the benefit of all," the statement said.