The incident happened about 5 a.m. Thursday on Housing Lane in Morrow.
Dashon Lee Davis, 30, turned himself in about 7:30 p.m.
He is charged with three counts of attempted first degree murder as well as the arson count.
He allegedly stabbed the woman several times in the neck and chest with a kitchen knife.
"After inflicting the wounds to the victim, Davis then intentionally set fire to the victim's bedroom. Davis also lit an additional fire in the hallway of the apartment/duplex," said Information Officer Jimmy Darbonne.
He said two children, ages one and five, were in the home at the time a fight broke out between the woman and Davis.
Neighbors forced themselves into the apartment and rescued the woman and children.
Darbonne does not know the relationship between the woman and Davis.