Fire crews gave the baby oxygen at the scene.
Police today confirmed the blaze was being treated as suspicious. Neighbours said the flats had been targeted before.
As she recovered at home last night, Oliwia’s father, who wanted to be identified only as Rafal, said: “I was downstairs in the garage and I heard something like a boom, and there was smoke. There was smoke in the house and the fireman saw we had a two-month-old baby, so to be safe they gave her oxygen.
“It looks like someone’s started it. The car was here for a long time, it’s got no tax, people have called the council about it – nobody uses the car.”
Neighbour Lyubomir Tonchev told how he had raced to wake his wife and young son after discovering the fire as he arrived home from a night shift.
The Bulgarian, who moved into the building less than a month ago, said: “I’d finished work and was trying to park my car when I saw the fire under the ground floor where the car park is so I called the police. And then I ran to get my wife and my baby out.
“It could have been much, much worse.
“The people that did this are really stupid. It’s dangerous.”
The fire started at around 6am yesterday in a Toyota in a private, secure car park under the flats near Hibs’ stadium.
The blue and silver vehicle was a burnt-out wreck and a door leading to the car park appeared to have been kicked in.
Another resident, named only as Lesley, said: “It looks like they smashed the window then poured lighter fuel into the car. I think the people that did this are scum. I feel sorry for the parents of the injured baby. There’s quite a few babies in here.”
She added: “It’s not the first time something has been set fire to in there. The CCTV doesn’t work.
“It’s shocking that people keep coming in and setting fires. It’s a danger to the children and for their parents.
“I got attacked in the car park. It’s quite a problem.”
Design engineer David Currie, 24, said he and his flatmates were woken at around 6.30am by firefighters.
He added: “It apparently happened in April 2008 as well, it was a lot more serious and everybody from the block had to be evacuated for 24 hours. It is a bit of a worry.”
A spokeswoman for Lothian and Borders Fire and Rescue said: “We attended a car alight in an underground car park in Hawkshill Close at 5.58am.
“We called an ambulance as a precautionary measure for a two-month-old baby suffering from smoke inhalation.
“Fire crews treated the child with oxygen at the scene.”