The song, from the 1939 film "The Wizard of Oz", is vying for the No. 1 slot in the UK's weekly list of the top 40 best-selling singles that are usually played in full on a BBC Radio 1 chart show on Sunday.
As pressure mounted from some politicians and members of the public to ban the "disrespectful" song, the BBC said it would air the track on Sunday but only a short clip as part of a "news item" explaining why the song is in the charts.
"The BBC finds this campaign distasteful but does not believe the record should be banned," the BBC said in a statement.
The furore was seen as a critical test for the new head of the BBC, Tony Hall, appointed after a sex scandal involving the late TV presenter Jimmy Savile threw the broadcaster into turmoil and raised questions about the organisation's ethics.
The publicly-funded BBC was criticised for a slow and unconvincing response to revelations that Savile was a serial paedophile abusing children over decades.
BBC Radio 1 controller Ben Cooper acknowledged the decision to air a clip of the Wicked Witch song was a compromise.
"You have very difficult and emotional arguments on both sides of the fence and let's not forget you also have a family that is grieving for a loved one who has yet to be buried," he said in a statement.
But the reaction not to play the song in full like other chart hits sparked criticism of the BBC caving in to political pressure over the 87-year-old leader who died of a stroke last Monday.
Even some Thatcher supporters, including the former Conservative MP Louise Mensch, said the BBC should air the song.
"Thatcher stood for freedom," tweeted Mensch.
Media lawyer Mark Stephens from London law firm HowardKennedyFsi said the BBC should play the song.
"She was a controversial figure and its chart position represents the views of a section of the community," he told the Independent newspaper.
Prime Minister David Cameron steered clear of the debate on Friday, having struggled all week to find a unified approach towards handling Thatcher's death and her funeral on Wednesday.
Earlier in the week he described so-called death parties by her detractors as "distasteful" and a spokesman on Friday said he had no further comment to add over the Wicked Witch song.
Deputy Prime Minister Nick Clegg, leader of the Liberal Democrat Party, said it was not for politicians to tell the BBC what to play even though he did not feel the online campaign was "in very good taste at all".
The BBC has banned other songs in the past, including the Sex Pistols' "God Save The Queen" which was released in 1977 at the time of the Queen Elizabeth's Silver Jubilee.
