The incident happened at around 04:15 at an address in Kenilworth Drive, Liberton. Nobody was hurt.
Officers said they are looking for a man who is described as being white and aged between 18-25. He was wearing a grey hooded top and grey trousers.
Det Insp Stuart Harkness said the nature of the incident was "serious and reckless".
"There will be additional officers patrolling the area in the coming days, and we will continue to work closely with the public in order to bring the perpetrators to justice," he said.
"We are appealing to members of the public or delivery drivers who may have been in the area at the time to get in touch if they witnessed any suspicious activity or vehicles."
Police have issued a description of a man they want to trace after a gun was fired in Edinburgh.