Moira was 11 when she vanished during a trip to the shops in Coatbridge, North Lanarkshire, in 1957.
Today a team of specially trained police personnel and forensic experts opened the grave at the town's Old Monkland Cemetery after claims her remains are there.
More than a dozen officers were working alongside forensic anthropologist Professor Sue Black at the site. The grave is known to contain eight bodies – but a radar examination suggests it may contain a ninth.
Three forensic tents were erected at the cemetery overnight. It is understood work to uncover the plot began in the early hours today.
A team forensic officers, carrying shovels and pick axes, arrived at the cemetery shortly before 10am today.
Mom-ents later, they were seen entering a tent covering the grave, carrying lighting equipment and telescopic ladders. Water was being pumped out of the sodden ground in preparation for work being carried out.
Chief Inspector Kenny McLeod, area commander for Airdrie and Coatbridge, is leading the investigation.
He said: "Together with Professor Sue Black and a team from the University of Dundee, we will be carrying out a detailed examination and are working to bring this line of the investigation to a conclusion.
"Records show that there are the remains of eight people in the burial plot. We know who should be in the plot. However, we have information that the remains of Moira Anderson may well be in this burial plot.
"This is sensitive, difficult work, which poses significant challenges, including the weather but we will be here on site until the work is concluded. We expect to be here for several days."
Moira, who lived in Coatbridge, is believed to have been abducted and killed. 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 today 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.