To stop the bleeding, take a clean towel or cotton cloth and apply pressure on the mole. When the bleeding stops, clean it with mildly warm water. Stopping the bleeding also depends on the type of wound; minor cuts can be stopped by applying pressure. Big cuts will take time; while applying pressure, take the dog to a veterinarian. Bleeding from moles can happen if your dog has scratched or bitten the mole or a cut from sharp objects.