The gunshots were discharged at around 1.30am yesterday at the front door of the house at Cooneybeg, Rathpeacon.
The family was said to have been left badly shaken by the incident.
They were asleep in their beds when the shooting occurred.
By the time gardaí arrived, the gunman had fled and efforts to identify him were continuing yesterday.
The scene was sealed off by gardaí for a technical examination and forensic experts started to comb the area after first light.
Gardaí are awaiting the results of ballistic tests to determine what kind of weapon was used in the attack.
It is believed that up to seven shots were fired at the front door, possibly from a handgun.
It was unclear if the gunman escaped in a getaway car driven by an accomplice, or was acting alone.
Gardaí have not released a motive for the attack and are said to be working on several lines of inquiry.
Gardaí are appealing for witnesses to the shooting at the house which is 7kms north of the city centre.
They are anxious to speak to anybody who was in the general area between 1am and 2am yesterday and who may have seen or heard any suspicious activity.
Anybody with information on the incident is asked to contact Mayfield Garda Station on 021 4558510.