About 11.20 pm on Sunday a black Vauxhall Astra parked in Hillview Crescent in the town was found to be on fire.
The owner extinguished the blaze but the car has been left badly damaged.
Enquiries are still at an early stage but officers believe the fire may have been caused by a firework being set off inside the car.
Constable Ross Hair said: “We have reports of a loud explosion at the car around this time.
“The consequences of this reckless act could have been so much worse had the fire not been tackled at an early stage.
“We are carrying out door to door enquiries in the area and I would ask that anyone who may have seen or heard anything suspicious in the Hillview Crescent area around this time get in touch with us at Annan on the 101 number, quoting the reference 4460 3/11.
“Anyone who wishes to pass on information and remain anonymous can do so by calling Crimestoppers on 0800 555 111.”
