The fentanyl mentioned in the GP (probably also what I had) is an anesthetic shot.
Still, I don't think the procedure used for wisdom teeth in the US has the same risks as the general anesthesia used in more involved surgeries. It's probably still reasonably described as sedation (in my story, I'm obviously still conscious, just not very aware. I didn't include it above, but I actually asked How much longer? and they just told me to go back to sleep).