Eight police cars and a dog unit were called to the scene in Polwarth at around 4am on Saturday morning.
Two people, understood to be the occupants of the stolen car, were taken to Edinburgh Royal Infirmary by ambulance.
Police said the incident began when the car was stolen elsewhere in the city and ended when it hit parked cars in Polwarth Gardens, Polwarth Grove and Polwarth Terrace.
There had been some suggestion from residents that police were pursuing the vehicle when it came to a halt.
Others said police cordoned off the street as the incident was dealt with. A large number of witness statements were taken, the police said.
Officers were today unable to confirm whether any arrests had been made in connection with the incident.
A spokesman for the Scottish Ambulance Service confirmed they had attended the incident.
He said: “We got the call from police at around 5.11am to a road accident on Polwarth Gardens.
“Two ambulances attended and took two people to hospital.”
Polwarth resident Ian Barclay, who lives near to where the incident happened, said locals were shocked to see the damage the following morning.
“I don’t think many people were aware of it happening at the time, certainly not where I live,” said the shop worker.
“But you could see quite a bit of damage the next day, I saw quite a lot of glass on the road.
“It’s unusual to see police activity around here, certainly when it comes to cordons and so on.
“Generally it’s very quiet so sirens coming through at five in the morning will have been quite a shock to a few people, I would have thought.”
A police spokesman said: “Reports came in about a number of vehicles being damaged in the Polwarth area.
“A number of cars were in attendance because of the high number of witness statements that had to be taken.
“It seems the incident did not start in Polwarth but ended there.”
• Meanwhile, an elderly man has been robbed of more than £100 as he stood at a cash machine in the Capital.
The incident took place at the Royal Bank of Scotland ATM on Nicholson Street at around 1.20pm on Saturday.
The victim is an 85-year-old, and a man is said to have run up to him and snatched the money from his hand.
Police are appealing for witnesses. Anyone with information should call police on 0131 311 3131 or Crimestoppers anonymously on 0800 555 111.