Moira was 11 when she disappeared during a trip to the shops in Coatbridge, North Lanarkshire, in 1957.
It is believed Moira was abducted and killed, but her body has never been found.
Forensic experts started excavating the plot at the town's Old Monkland Cemetery yesterday, to see if the schoolgirl's remains were hidden there.
A large section of the cemetery has been cordoned off with police tape, with a large black tent and two smaller blue tents erected around the grave.
The excavation is being led by Professor Sue Black and a team from the forensic anthropology department at Dundee University.
As reported in later editions of last night's Evening Times, police believe there may be up to eight bodies interred in the burial plot.
Chief Inspector Kenny McLeod, area commander for Airdrie and Coatbridge, said: "We may not need to exhume all eight human remains, but that does remain a possibility.
"The site presents significant problems and challenges, not only in terms of ground conditions but also record-keeping, as you can imagine, going back to 1908, it may not detail every person in the burial plot.
"This is sensitive, difficult work but we will be here on site until it is concluded. We expect to be here for several days."
Officers stood down at 5pm yesterday, and work will continue today.
Bus driver Alexander Gartshore, who died in 2006, has been linked to Moira's disappearance.
It is believed her body was hidden beneath the coffin of a man, Sinclair Upton, who died a few weeks after she went missing.
The grave opened yesterday contains the body of Mr Upton, an acquaintance of Gartshore.
Gartshore's daughter Sandra Brown, also from Coatbridge, believes her father was responsible for Moira's death.
Sandra, a former Evening Times' Scotswoman of the Year, also said she believes Moira's body was hidden in the burial plot.
Chief Inspector McLeod added: "This is a very sensitive time for not only Moira's family, but the families of those who are buried in this plot.
"Any human remains we discover will be treated with dignity. We can't lose sight of the real human story here.
"Moira Anderson went missing in 1957 and we have a responsibility to her family to bring this investigation to a conclusion."
Experts are exploring the possibility that Moira's body may have been dumped under a coffin in the grave, which was thought to have been open around the time of her disappearance.
The operation comes after a sheriff gave police the go-ahead last month.