The duo had darkened their skin and sported prosthetics to go undercover and prank celebrities on the ITV show.
In a statement on Twitter, 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steps to ensure footage was taken down and have again recently confirmed with ITV that these segments, and any other historical content that could cause offence, does not appear on either the ITV Hub or the Saturday Night Takeaway YouTube Channel."
In 2003, the pair dressed as Jamaican women to surprise Emmerdale stars, and the following year they impersonated Japanese girls, using make-up and exaggerated accents.
They are the latest TV stars to apologise for causing offence, amid increased awareness of the Black Lives Matter movement following the death of George Floyd, who was killed by police in the US while being detained.
Comedian Leigh Francis last week expressed his remorse for portraying black celebrities on Bo' Selecta!
Little Britain was this week removed from iPlayer over the use of blackface in some sketches.