Ismail Mohamed Abdulwahab from Brixton, south London, died in hospital in the early hours of Monday.
He had tested positive for Covid-19 on Friday, a day after he was admitted to King's College Hospital, his family said.
Family friend Mark Stephenson said the boy's mother and six siblings are now awaiting the results of a post mortem.
A spokesman for King's College Hospital NHS Foundation Trust said: "Sadly, a 13-year old boy who tested positive for Covid-19 has passed away, and our thoughts and condolences are with the family at this time.
"The death has been referred to the coroner and no further comment will be made."
Following the release of the latest official figures, a 19-year-old was believed to be England's youngest victim to have died in hospital with no existing medical issues.
Ismail's family said they were "beyond devastated" by his death, in a statement released through Mr Stephenson.
"Ismail started showing symptoms and had difficulties breathing and was admitted to King's College Hospital," the family said.
"He was put on a ventilator and then put into an induced coma but sadly died yesterday morning. To our knowledge he had no underlying health conditions. We are beyond devastated."
Mr Stephenson, college director at the Madinah College where Ismail's sister works, has set up a GoFundMe page to raise money for funeral costs.
The page says: "Sadly he died without any family members close by due to the highly infectious nature of Covid-19."