State laws vary but in general if the person with the registered and insured vehicle is at-fault in the accident than yes, normally they will be held responsible and his insurance would accept a claim by the owner of the unregistered vehicle.

While the car may have been illegally on the roadway due to expired registration or total lack of registration on it, that fact alone does not make the person at fault or keep the victim (person hit) from being able to make a claim with the at-fault party.

If police were called to the scene of the accident hopefully the person with the unregistered was cited so that they will face the penalties the state has in place for not having proper registration on their vehicle and driving on the public roadway.