The TV stars wore dark make-up and prosthetics for the Undercover sketches, in which they pranked other celebrities while in disguise.
In a joint statement, the presenting pair said they would not make those sketches today and had requested ITV, which broadcasts the show, remove them from their streaming catch-up service and YouTube channel.
They said: “During past episodes of Saturday Night Takeaway we impersonated people of colour in the Undercover segment of the show.
“We realise that this was wrong and want to say that we are sincerely sorry to everyone that we offended.
“We purposely stopped doing this several years ago and certainly would not make these sketches today.
“We had already taken steps to ensure footage was taken down and have again recently confirmed with ITV that these segments, and any other historical content that could cause offence, does not appear on either the ITV Hub or the Saturday Night Takeaway YouTube Channel.”
The pair used blackface in 2003 as they dressed up as two fictional Jamaican women, Patty and Bernice, to prank Emmerdale cast members.
And in 2004 they dressed up as two Japanese girls, Suki and Keiko, using make-up and exaggerated accents.
Their apology follows widespread Black Lives Matter protests, in the US and the UK, after the killing of George Floyd.
Earlier this week Little Britain was removed from BBC iPlayer after coming under fire because of the use of blackface in some sketches.
And last week, comedian Leigh Francis issued a tearful apology for portraying Black celebrities on sketch show Bo’ Selecta.