The family woke up to the sound of an explosion outside their home in Clayton Crescent, Waterthorpe, to find a carÂ engulfed in flames outside.
Firefighters raced to the scene with fears that the flames could have spread to the house.
Marie Burns-Jenson, whose son's hire car was set alight, said it wasÂ parked outside her neighbour's home while he is visiting with his family from Australia.
'We were in bed when we heard a car alarm sounding and my husband looked out of the window and the car was fully ablaze right outside a neighbour's house.
'My neighbour and her family heard a huge bang when the car went up and she said she thought a car had gone into her house at first. They had to evacuate. It must have been terrifying.'
She saidÂ footage from a CCTV camera at her home shows a figure walking towards the car before it went up in flames.
The car, which was totally destroyed in the arson attack, has been towed away.
Mrs Benson-Jenson added: 'Everyone is shaken up, nobody can understand why somebody would do this.
'We don't know if the person deliberately choseÂ this car or if they got the wrong one.
'It is a close knit community round here, we don't get anything like this happening. It is bizarre.'
Anyone with information should call South Yorkshire Police on 101.