Marilyn Gregg, who lives in a Housing Executive house in Somerset Street, reported a rat problem in February and was visited by pest control from Belfast City Council.
But the dead rats have been left to decompose on her property.
The Housing Executive said it was providing deodoriser as a "courtesy".
Ms Gregg told BBC News: "The smell is rotten.  I haven't been living here since February because the smell is so disgusting.  It's been almost four weeks and the smell is still very very strong.
"The Housing Executive are my landlords, why should I have to fix this problem? I don't think the best has been done for me.
"I'm not coming back until this is resolved.  I can't stand the smell."
Ms Gregg said she has been in touch with her landlord a number of times.
A council spokeswoman confirmed its pest control team was called out by the Housing Executive.
"As the Housing Executive is the landlord for this property, it would be their responsibility to take any further action needed regarding the removal of dead rats at this address," she added.
A Housing Executive spokesperson said the pest controller had explained to the tenant that, in rare cases, rats may die in the cavity under a property once poison has been applied.
"The tenant was aware this was a possibility and this now appears to have happened," the housing authority added.
"Again, we sought advice from pest controllers and, while it is not our responsibility to deal with this issue, as a courtesy, we are locating and providing large deodoriser bags to be placed under kitchen units to absorb any unnecessary odours.
"Meanwhile, contractors have been engaged to carry out remedial works at the property to ensure pests cannot return."
Deodoriser bags should deal with the smell for the four weeks it is expected to take for the rats to decompose.
A Belfast woman who has had to leave her home because of the smell of dead rats says she feels like a second-class citizen.