BBC Radio Leeds ran the feature during its Sunday breakfast programme, asking listeners to try and guess the famous person from a series of songs.
But listeners were angered when it became clear guest host Nathan Turvey had chosen serial killer Brady, playing Psycho Killer, and the theme tune to the Brady Bunch.
He also played All the Young Dudes and Suffer Little Children by The Smiths, a track written about the Moors murders.
The BBC has since apologised for the “unacceptable” feature.
Twitter users were left amazed by the broadcast.
Stuart Petch branded it "crass to the max", writing: "Think that's funny? You're kidding me?”
Mr P said: “Please tell me that this isn’t genuine.”
Darren Longhurst PT said: “Jeez....someone left their brain cell on the bus #epicfail”
Brady and his partner Myra Hindley tortured and murdered five children in the 1960s, burying their bodies in Saddleworth Moor.
He died last week having failed to reveal the burial site of Keith Bennett, one of the victims to the couple’s horrific killings.
BBC Radio Leeds ran a feature playing four songs asking listeners to guess who the songs were related to.
Announcing the winner Mr Turvey said: “I think most of you worked out the answer was Ian Brady, which is someone we’re talking about this morning of course, who died earlier in the week.”
A spokeswoman for the BBC said: “This is clearly unacceptable and we apologise."
Police in the north west have vowed never to stop searching for the remains of 12-year-old victim Keith Bennett, pledging to act on all “credible and actionable information”.
Moors Murderer Ian Brady - In pictures
