A 48-year-old woman from the same address in Danube Street was also taken to hospital but her injuries were not said to be life-threatening.
It was reported that the girl had suffered stab wounds.
Police said they were not looking for anyone else in connection with the incident at about 01:35 on Wednesday.
A spokesman for Police Scotland said: “Police in Edinburgh were called to an address in Danube Street at around 01:35 on Wednesday 7 December.
“An eight-year-old girl was found with serious injuries and was taken to the Royal Hospital for Sick children.
“A 48-year-old woman was also taken to Edinburgh Royal Infirmary with serious but non-life threatening injuries.
“Enquiries are ongoing into the full circumstances. However, officers are not currently looking for anyone else in connection with this incident.”