Two pipe bombs exploded at Mill Street, Lambeg, at around 2.20am on Tuesday.
Alderman Stephen Martin said: “It is devastating to hear of an attack on the home of a young mother, who was in the house with her child when the incident occurred.
“We are extremely lucky we are not facing a more severe situation this morning and after being on the scene and speaking to local residents, I want to praise the quick and efficient effort of all involved, which not only saw a third device defused, but also the safe evacuation of those living close by.
“Everyone must have the right to live peacefully in their own home, without the threat of harm, and this type of behaviour is not acceptable in our Borough. I would urge anyone with information to contact the PSNI as soon as possible, so that those involved can be dealt with before the courts.”