John Docherty, 49, was yesterday convicted of the murder of Elaine Doyle, who was killed in Greenock, Inverclyde, in June 1986 while she was walking home after a night out.
Police finally tracked down Docherty after traces of his DNA were found on tapings which had originally been taken from the victim's body in the hope of extracting hair or clothing threads.
The majority verdict, at the High Court in Edinburgh, ended the near three-decade wait for justice endured by Ms Doyle's family. The jury rejected 49-year-old Docherty's claim he had been at home with his parents when the girl was attacked, stripped naked and strangled yards from her home.
Docherty had lodged a special defence of incrimination, claiming the culprit may be among a list of 41 names from files of the police investigation.
Ms Doyle's older brother, 49-year-old John, was in court to hear the jury return their verdict after about four hours of deliberation. He said the jurors had made the "right decision".
His father, Jack, died in 2012 after repeatedly campaigning for his daughter's killer to be brought to justice and his mother Maureen was said to be too ill to go to court.
In a statement, she expressed her thanks to police officers who had worked on the case, Lord Advocate Frank Mulholland and trial prosecutor John Scullion, for securing the conviction.
She also said she was grateful to the people of Inverclyde "who have lived this nightmare with us for the last 28 years and have always supported our family".
She added: "The result at court doesn't make our day-to-day living any easier - the pain doesn't go away. But my son John and I take comfort we now have justice for Elaine, which is all we, especially her dad Jack, campaigned for."
Docherty was identified after police reviewing the case looked at a witness statement from a friend, taken shortly after the murder, which stated that he had been in the same club as Ms Doyle on the night she died.
Detectives had obtained a full unidentified DNA profile as part of a forensic re-examination of the case in 2005, which they matched to Docherty after they tracked him down and obtained a voluntary sample in mid-2012.
The trial heard the odds of the DNA belonging to anyone else were one billion to one and the spot where it was deposited fitted the scenario of an assailant strangling the victim from behind.
Detective Superintendent Bobby Hendren, the senior investigating officer in the case, paid tribute to the Scottish Police Authority's forensic scientists, whose support he said had been vital in bringing Docherty to justice.
He added: "This investigation has always been about Elaine Doyle and her family and bringing the man responsible for her murder to trial.
"I would like to pay tribute to her mum and brother who have continued to support us throughout this investigation and I am sorry that her dad is no longer with us to hear the verdict. I can only hope that today's verdict brings them some comfort. I would also like to thank the people of Greenock for their continued help and support in the past 28 years.
"I, and all the officers who worked on this investigation, are well aware that the senseless murder of Elaine has cast a long shadow over the town of Greenock and we would not be where we are today without their support."
Ms Doyle's body was found on the morning of June 2, 1986. She was lying on her front, with a mark round her neck, after a struggle experts believe might have lasted minutes. A clump of hair had been torn from her scalp.
A police investigation was launched but no strong suspects were identified until the DNA match with Docherty, a former Royal Engineer who had returned to Greenock after a spell in the Army and later worked as a driver with Inverclyde Council, was obtained.
Another link in the chain came when a key witness, Martin Brown, 52, looked at a photos taken to his London home. Mr Brown described seeing a man with staring, angry eyes and dark clothing apparently following Ms Doyle on the night she died. He told detectives in September 2012 a photo, culled from Docherty's Army ID card, reminded him of the fleeting encounter 26 years earlier.
Judge Lord Stewart told jurors that although they had heard much criticism of the early stages of the investigation, it should be put on record that "extraordinary persistence, skill and determination" had brought the case to court.
The judge told Docherty the only sentence for murder was life, but called for a background report before fixing the minimum sentence he must serve before he can apply for parole.
Docherty - who had been on bail - was remanded in custody to await formal sentencing in Glasgow in August.