It is understood the shots were fired from a high powered ball bearing gun.
A statement released by the party on Sunday said the front windows and door of the councillor's house were struck seven times.
The incident is believed to have happened sometime between Thursday night and Friday morning.
The councillor, her husband and baby daughter were not at home at the time.
Ms Hanna, the Chair of the East Belfast Policing and Community Safety Partnership, said the attack disgusted her and showed utter contempt for her young family.
"It's a violation of our security," she told UTV.
"My husband and I have a young daughter and obviously, her safety and security is our principle concern in life and this incident has very much shaken us."
The last number of weeks have been disturbing for those of us who aspire to a genuinely shared future, and to the principle that all here can be British, Irish or both as they so choose.
Ms Hanna said she would not be intimidated by what she described as fascist elements with no respect for democracy.
"This attack won't stop me working to promote core SDLP values of partnership, equality and mutual respect."
South Belfast Alliance MLA Anna Lo said there was "no justification" for attacking the home of a hard working elected representative.
"How many more attacks on elected representatives, their staff and the PSNI will we have to suffer until Unionist politicians realise that the illegal protests they are continuing to attend are doing nothing but increasing fear and tension in the community," she added.
Meawhile, UUP leader Mike Nesbitt said: "I join with others in condemning the actions of cowards in attacking the Hanna family home.
"It is another pointless act of violence".