Ciaran James Williamson died after the incident at Craigton Cemetery, in Cardonald, at about 19:25 on Tuesday.
He was pronounced dead at Yorkhill Hospital, where one of the 10-year-olds is in a stable condition. The third boy was later released from hospital.
Police said they were not not treating the incident as suspicious.
In a short statement issued by police, Ciaran's family said: "We would like to take this opportunity to inform you that our son Ciaran James Williamson sadly passed away in a tragic accident at Craigton Crematorium.
"We would respectfully ask that you respect our privacy at this distressing time."
Det Insp Tom McKean, of Police Scotland, said: "This is such a tragic set of circumstances and our thoughts are with Ciaran's family.
"Inquiries are ongoing, however the incident is not being treated as suspicious and a full report will be submitted to the procurator fiscal."
An eight-year-old boy has died and two 10-year-olds have been injured in a "tragic accident" involving a falling headstone at a cemetery in Glasgow.