The car had been left on Park Street South in Blakenhall for around a week after it had run out of petrol.
The woman returned today to put fuel the vehicle, but the front of the car caught fire as she tried to start it.
The woman or her child were not hurt.
Firefighter Will Lane said: "The cause of fire has gone down as petrol on a warm surface, probably the engine.
"Neither of them were injured."
A worker at a nearby factory, who did not want to be named, said: "My car was parked behind her car.
"She got in and started it but it just went up. There were a few people who had gone out to help her. They took her baby out of the car and took them both next door to warm up.
"The fire service arrived in about five minutes and put it out so it was all over quite quickly."