There are several options that aren't narcotic based when it comes to pain relief. The old over the counter , non steriodal, anti inflammatories do wonders for pain from inflammation ( dental pain is one of those) and taken as directed, they can be the best for that type of pain relief. They also work really well when it comes to any pain that is caused due to inflammation.
As far as other types of pain, lidocaine works for topical pain, some of the old tricyclic anti depressants work really well. Lyrica, gabapentin and several other seizure type of medications work for neuropathic pain and cymbalta has recently been added to that list.
Your doctors are the best sources for ideas to manage pain that may come and offer suggestions for treating it without the use of narcotics. They have many options available to share with you.