The news-related feature aired on BBC Radio Leeds during Sunday’s breakfast show, with listeners being asked to guess the identity of the person using four songs as clues.
DJ Nathan Turvey, who was standing in for Radio Leeds’s regular Sunday presenter, played Mott The Hoople’s All The Young Dudes and Brady Bunch theme tune, followed by The Smiths’ Suffer Little Children, which was written about the Moors murders
The last song played was the Talking Heads track Psycho Killer.
After playing the tracks, Turvey could he heard saying: “They were four songs all linked to someone in the news this week, all you’ve got to do is work out who it is.”
Turvey later played the montage again and 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.”
The material triggered a social media backlash after it was posted on Radiofail, a website featuring mistakes by DJs.
Broadcaster Julia Hartley-Brewer wrote on Twitter: “Oh. My. God. What was BBC Radio Leeds even thinking?”
Also on Twitter, Sofia Cann wrote: “I thought the Beeb had/have producers? It must have passed a few people before us lot got to hear it.”
A BBC spokesman said: “This was clearly unacceptable and we apologise.”
Child-killer Brady died last week aged 79 after spending half a century in prison for his crimes.