anaesthetic

administer, give sb

The dentist will administer a local anaesthetic before removing your tooth.

use

The dentist used a strong anaesthetic before removing my wisdom tooth.

have

The patient will have a general anaesthetic before the surgery tomorrow morning.

take effect, work

The local anaesthetic worked quickly, and the dentist began the procedure within minutes.

under (an/the) ~

The dentist said my tooth extraction would need to be done under anaesthetic.

without (an/the) ~

The patient was conscious throughout the entire procedure because the dentist administered the injection without anaesthetic.