Moira disappeared almost 56 years ago, when she was 11, and detectives believe she may be buried in a plot in Old Monkland Cemetery in Coatbridge, North Lanarkshire.
Yesterday, just as the light was coming up and the M8 rush hour traffic was getting heavier, forensic anthropologist Sue Black, of Dundee University, and her team of three started to sift through the earth.
They face several challenges in the job: the site is waterlogged and the coffins in the family plot – there are eight in all – will have disintegrated.
However, there is reason to believe Moira may be there. Records show there are eight bodies buried in the plot, but a radar examination has suggested there may be a ninth.
The work in the cemetery is going ahead after Moira's sisters Janet Hart and Marjory Muir went to Airdrie Sheriff Court to seek permission last month.
Moira disappeared in February 1957 after going out on an errand to a shop in Coatbridge.
The prime suspect for her murder was Alexander Gartshore who died in 2006.
Gartshore's daughter, Sandra Brown, who visited the cemetery yesterday afternoon, has said she believes Moira's body was hidden beneath the coffin of a man called Sinclair Upton, one of the eight people buried in the plot.
A section of the cemetery has been cordoned off with police tape. A large black tent and a series of smaller blue tents have been erected around the excavation site.
Officers in overalls went in and out of the tents as work started.
Chief Inspector Kenny McLeod, area commander for Airdrie and Coatbridge, said the search of the plot would take several days.
He explained: "What we're going to do today is a detailed examination of a burial plot, which is part of the long-term missing person's inquiry dating back to 1957.
"There are significant challenges for Sue Black and her team, including ground conditions and record keeping. We can't guarantee what human remains will be in there and that's the reason we're here.
"We have a real responsibility to Moira, but to her family as well. It's important her family get closure and have a conclusion to her story."
Mr McLeod added: "It is a sensitive operation. We have to deal with any human remains with dignity. The forensic anthropologists can identify the age and sex of any skeleton or human remains."
Asked why the application to exhume the grave had come from Mrs Brown's charity the Moira Anderson Foundation and not sooner from the police, Mr McLeod said: "The police follow any line of inquiry – we have worked in conjunction with the Moira Anderson Foundation and were informed last year they were placing a petition before the court.
"We want to bring a conclusion to this line of inquiry. The foundation should be commended for their tireless dedication to this inquiry."
The case was reviewed in 1983, and again in 2003. It is currently undergoing another review.
Gartshore was named as the killer in a deathbed confession by a fellow child abuser.
Mrs Brown has investigated the case for years and wrote a book on the subject in which she said her father almost certainly murdered Moira.
At the end of the last year, Moira's sister Janet Hart said she hoped the exhumation would bring the case to a close.
"It's not too much to ask that my little sister Moira have her final resting place, not in an unmarked grave or in someone else's grave but with her loving parents, who grieved for their lost daughter daily until they too were laid to rest," she said.
Mr McLeod added: "Operations like this are not taken lightly as we remain sensitive to Moira Anderson's family and also the families who have relatives buried elsewhere in the cemetery."
Strathclyde Police said officers stood down at 5pm, with work continuing tomorrow.