This is a mistranslation. There is no pronoun in the Greek text, no "their" which would have to be genitive. The word is the definite article, and it is in the dative case, as is the word "dogs." The correct phrase is "τοις κυναριοις" = "to the dogs." And actually, it is not "dogs," it is "puppies."
"Yes, Lord," she said, "but even the dogs eat the crumbs that fall from their masters' table."