The blaze was discovered at the rear of a house in Dalswinton Avenue at about 12.20 am on Wednesday.
Officers say it appears that a sofa and some rubbish had been set alight. The fire was extinguished by the Fire and Rescue Service but not before it had damaged a window in the unoccupied house.
Constable Richard Davidson at Dumfries said: “This appears to have been a complete act of recklessness and the consequences could have been tragic if the fire had taken hold. We are not sure at this stage if those responsible knew that the house was unoccupied at the time the fire was set.
“We want to hear from anyone who may have any information which might help us catch those responsible for this stupid and dangerous act.”