A petrol bomb was thrown at the front window of the house in Wallace Park at 0345 BST on Sunday. The flames were put out by a passer-by.
A number of people inside escaped injury. Police said a sectarian motive was one line of enquiry.
Later on Sunday, a bullet was pushed through another family's letterbox.
That incident - on a Protestant family - is also being treated as sectarian. A brick was also thrown at their house.
Rasharkin is a religiously mixed village, though with a Catholic majority.