If a patient is unresponsive and bleeding from the mouth, what is the best course of action to control the bleeding

Confirm that the bleeding is actually from within the mouth and not from the gastrointestinal tract. If blood is coming up from the throat or being vomited, there could be a serious medical problem...

If the patient is not responsive, the rescuer should lay the patient in the recovery position and suction the mouth to prevent the airway from becoming blocked.