In December, the rock band appeared on 5 Live over a live link from the US to discuss the Facebook campaign promoting their track 'Killing In The Name' as a rival to The X Factor's Joe McElderry for the Christmas Number One.
Following an interview at 9am, the band performed the song live, including frontman Zach De La Rocha uttering the line, "F**k you, I won't do what you tell me", four times before the feed was faded out.
Show presenter Shelagh Fogarty then said: "Sorry, we needed to get rid of that because that suddenly turned into something we were not, well we were expecting it and asked them not to do it, but they did it anyway - so buy Joe's record."
After the transmission, a viewer complained to Ofcom that the live performance was offensive and inappropriate for a breakfast show.
The BBC accepted that the language used was "neither appropriate nor justified on a morning programme on Radio 5 Live".
However, the corporation stressed that Fogarty had immediately apologised for the utterance and her co-presenter Nicky Campbell repeated the apology later in the programme.
An additional apology was placed by the programme editor on his blog and issued to the 32 people who complained directly to the BBC.
The 5 Live team said that it was aware of the song's "very strong language", and had highlighted to the band on three separate occasions about the need not to use the original lyrics during the live performance.
The BBC acknowledged that there was a risk in permitting the live performance, but it judged that "there was considerable public interest in the battle for the Christmas Number One and it was felt that a rare opportunity to hear the band play live was editorially justified".
In its ruling, Ofcom acknowledged the editorial justification for Rage Against The Machine appearing on 5 Live to discuss the Facebook campaign.
The media regulator also welcomed the BBC's efforts to stop the band from using strong language during the performance and its subsequent apologies for the occurrence.
"Ofcom considered, however, that the language was likely to have gone beyond the expectations of the audience for a programme of this type and at this time. It was concerned that the programme's producers were well aware in advance that the original lyrics contained very strong language," said Ofcom.
"In addition, the very nature of the song was about refusing to conform to society's expectations, as suggested through the lyrics, "F**k you, I won't do what you tell me". Yet despite this, the band's singer was able to repeat the lyrics, "F**k you, I won't do what you tell me", four times before the song was faded out by the producer. This last point is of particular concern in view of the fact that the producers had full control over the output since it was provided over a live feed from the United States.
"However, given the measures taken by and assurances given to the broadcaster before the broadcast, the conduct of the band during the interview and start of the song performance, and the apologies issued, we consider that on balance this particular case should be resolved."