A good cleric would have DR x/Evil (since it would take an Evil weapon to overcome it).
A creature with an alignment subtype (chaotic, evil, good, or lawful) can overcome this type of damage reduction with its natural weapons and weapons it wields as if the weapons or natural weapons had an alignment (or alignments) that match the subtype(s) of the creature.
So... the answer is no, unless the dragon has the evil subtype (most evil dragons don't).