Ms Pirrie, who is understood to have a young son, was treated at the scene but later died in hospital.
Police Scotland said a force vehicle was following a silver Ford Ka after it failed to stop for officers.
The police car’s blue lights and siren were being used in the pursuit, the force said.
Three males have been detained over the crash and officers are searching for a fourth. Their ages have not been specified.
The Police Investigations and Review Commissioner (Pirc) is also reviewing the circumstances that led to the death.
Chief Superintendent Kenny Macdonald said: “Our sincere condolences are with the family and friends of the woman at this tragic time, and we will utilise all resources at our disposal to investigate this collision.”
He added: “We reported this collision to the Crown Office and Procurator Fiscal Service, who have instructed Pirc to carry out an independent investigation.
“Pirc are instructed to investigate deaths following police contact or involvement, and we will co-operate fully with their inquiry.”
A Pirc spokeswoman said: ”A police vehicle was following a silver Ford Ka when the Ka was involved in a collision which resulted in the death of a 33-year-old woman in Edinburgh.
”The Crown Office and Procurator Fiscal Service has instructed the Pirc to carry out an independent investigation into the circumstances leading up to the incident.
”Pirc investigators attended the scene in the early hours of the morning and a report will be submitted to the Crown Office in due course.” Anyone with information can contact police on 101 or anonymously via Crimestoppers on 0800 555 111.