Can dental anesthesia give you a headache?

Yes: A dental anesthetic can contribute to symptoms of a headache if the anesthetic is injected into a venous plexus or an artery. Patients with sensitivity to Epinephrine may also report headache type symptoms after receiving a local anesthetic containing epinephrine.

Possible: It's possible that an anesthetic triggers headache but it's more like that dental work itself could cause the headache.