Remains of the Moors Murderers' first victim were kept by police for 30 years without her family's knowledge.
Ian Brady and Myra Hindley admitted killing 16-year-old Pauline Reade in 1963, while in prison in the 1980s.
Greater Manchester Police (GMP) said it "recently became aware" some of her remains were kept "for investigative purposes" and has since apologised.
The teenager's niece told the Manchester Evening News she was "devastated" by the discovery.
Jackie Reade said: "The police called me in the middle of August and said they had 'stumbled' across some stuff.
"They said it was a piece of her jaw-bone and I couldn't believe it. It was heartbreaking.
"We thought she was all there together [...] These parts should not have been separated from her."
Pauline disappeared on 12 July 1963 while on her way to a disco near her Manchester home.
Her remains were found on Saddleworth Moor in Oldham in 1987 and she was buried at Gorton Cemetery.
Following Brady's death in May, an audit was carried out and some of her remains were discovered at the University of Leeds, where they had been kept on behalf of Greater Manchester Police (GMP).
- The 16-year-old was the first victim of Brady and Hindley, who attracted notoriety for kidnapping, torturing and killing five children and teenagers in the 1960s
- Hindley, who was acquainted with Pauline, lured the teenager to Saddleworth Moor with a fabricated tale about needing help to find some lost gloves
- It was there that the teenager was beaten about the head and her throat cut with such force that her spinal cord was severed
- It would be more than 20 years before Brady and Hindley admitted her murder and, in 1987, the pair led police to the site where Pauline had been buried in a shallow grave, still wearing her pink and gold party dress
- It took pathologists a month to decide she had been sexually assaulted
Pauline is buried is the same plot as her parents and brother. The remains held by police have been returned to the family.
Her niece said: "As a family we want to put all the parts back with Pauline, where they should be.
"But to do that we will have to disturb all four graves... We will hold second funerals for all of them."
Martin Bottomley, head of GMP's cold case unit, said: "The samples had originally been kept for investigative purposes.
"As soon as we became aware of this, we contacted Pauline's family to make arrangements so that the samples could be laid to rest in whichever way they felt most fitting.
"This is a deeply sensitive matter and we regret the understandable distress and upset this has caused... we have given them a number of options, all of which GMP will pay for."
Hindley died in prison in 2002 while Brady died at Ashworth Hospital, a secure psychiatric unit in Merseyside, in May.
His ashes were disposed of at sea on 26 October.