The family were woken by a loud bang outside the property in Northfield Park, Annan at around 00:15am on Monday and discovered that a BMW parked in the driveway was on fire.
Police said the blaze quickly spread to the house, which was completely destroyed by the flames, as was the car.
The three adults and three children – aged ten, four and nine months – escaped from the house and the fire brigade was called out.
Following a joint investigation with the Scottish Fire and Rescue Service, police are treating the fire as suspicious and are urging anyone with information about the incident in the Dumfries and Galloway town to contact them.
Detective Inspector Dean Little said: “The occupants of this house had a very lucky escape in this fire which has spread from the car to the house and destroyed both as a result.
“We are now conducting inquiries into what is a serious case of wilful fire-raising which could have had tragic consequences had the family not been able to escape the house.
“We are carrying out door-to-door inquiries in the area and will be reviewing CCTV which we have been able to retrieve from the community.
“I am now appealing to the community, particularly the local residents in and around Northfield Park or anybody who may have been in the area at around midnight on Sunday night, to come forward with any information which might help us in this inquiry.
“Did you see any persons or any unfamiliar vehicles in the area around this time? If you did I would urge you to contact us.
“I would also ask anyone who may have a private CCTV system to get in touch in order that we might review any footage which may assist in this inquiry.”
No-one was injured as a result of the fire.
Mr Little added: “Anyone who has any information which might help us trace those responsible for this horrendous crime is asked to call us at Annan on the 101 number, or if they want to remain anonymous then through Crimestoppers on 0800 555 111.”