The family of the 11-year-old, who vanished more than 55 years ago, had raised a legal action to exhume eight bodies from a lair where they believe she was secretly buried.
The application by Moira's sisters Janet Hart and Marjory Muir was yesterday given the go-ahead during a hearing at Airdrie Sheriff Court.
An investigation of the grave at Old Monkland Cemetery in Coatbridge, North Lanarkshire, will be undertaken by Strathclyde Police in the new year.
Speaking from Australia after Sheriff Frank Pieri's decision, Mrs Hart, 67, said: "I am over the moon that a decision has now been made and it is in our favour. It is such a sense of relief and hopefully this will give us closure after all these years."
Moira vanished on February 23, 1957, after being sent on an errand to the local Co-op near her home in Coatbridge.
The prime suspect for her murder was paedophile Alexander Gartshore.
Gartshore, who died in 2006, was named as Moira's killer in a deathbed confession by a fellow convicted child abuser.
Gartshore's daughter Sandra Brown wrote a book outlining how he almost certainly murdered the schoolgirl. Yesterday, she said: "There is nothing I have said about my father that is fantasy. It can never be right to take the life of a child and I do think this grave has secrets to give up."
The investigation of the site will involve the exhumation and reinterment of eight bodies. They are Elizabeth Upton, who died in 1908, Joseph Thom, who died in 1923, Margaret Upton, who died in 1951, Sinclair Upton, who died in 1957, Elizabeth McNeilly, who died in 1976, Peter McNeilly, who died in 1978, Hugh Winterbottom, who died in 1985 and Mary Winterbottom who died in 1995.
At the time of Moira's disappearance, Gartshore, was on bail after being arrested over the rape of a 13-year-old babysitter.
He was quizzed by police in the 1990s about Moira's death but denied any involvement.
David McKie, of Glasgow law firm Levy & McRae, the Anderson family's solicitor, said: "I am delighted the legal process has allowed the family to take the next step forward in seeking to find out what happened to Moira. It was a unique and complicated case and one which has taken a long time to resolve, but I hope the ultimate outcome proves it was worth the wait."