It comes after the broadcaster received almost 19,000 complaints about the incident.
The BBC had previously accepted that using the racial slur had caused offence, but had not apologised for it, saying the use of the N-word was “editorially justified given the context”.
But in a statement on Sunday, the BBC’s director-general Lord Tony Hall said: “Every organisation should be able to acknowledge when it has made a mistake. We made one here.”
In a email to all BBC staff, Hall wrote: “It should be clear that the BBC’s intention was to highlight an alleged racist attack. This is important journalism which the BBC should be reporting on and we will continue to do so.
“Yet despite these good intentions, I recognise that we have ended up creating distress amongst many people.
“The BBC now accepts that we should have taken a different approach at the time of broadcast and we are very sorry for that. We will now be strengthening our guidance on offensive language across our output.”
The N-word was used during a segment about a racist attack in Bristol.
During the segment, reporter Fiona Lamdin said: “Just to warn you, you are about to hear highly offensive language. Because as the man ran away they hurled racial abuse, calling him a n*****.”
The BBC’s decision to use the racial slur caused a huge backlash and saw Radio 1Xtra presenter Sideman – whose real name is David Whitely – quit the BBC “effective immediately”.
Whitely said the “action and the defence of the action feels like a slap in the face of our community”.
In an Instagram video, the DJ said: “This is an error in judgment where I can’t just smile with you through the process and act like everything is OK.
“I’m happy working with organisations until we all get it right, but this feels like more than getting it wrong.”