This came after complaints that the latest column - on coloured people - was racist.
Mbhele said Avusa Media and Sunday World acknowledged the outcry over the column, headlined "Jou ma se kinders".
He said it made derogatory generalisations about coloured people which were in clear violation of the South African Press Code and Avusa Media's internal codes. Mbhele said he took full responsibility "for the offending column".
"This column showed clear prejudice against a section of South African society. For this I unreservedly apologise to the South African society and to Sunday World readers," he said.
Avusa Media editor-in-chief Mondli Makhanya said the company had begun an internal inquiry into the matter.
Roberts' column comes amidst a furore over remarks made about coloureds by government spokesperson Jimmy Manyi.
Manyi, then director-general of labour, said in a show broadcast on KykNet's Robinson Regstreeks in March 2010 that there was an "over supply" of coloureds in the Western Cape.
Earlier yesterday, the SA Human Rights Commission said it was investigating a complaint about Roberts' column, spokesperson Vincent Moaga said.
City Press editor Ferial Haffajee said firing a columnist was not "the way to go".
"I think she should apologise and the paper should apologise and give its readers the right to reply."