You will start 2 different missions with the Anesthetic in your inventory, 'Anathema' and 'Invitation To A Party'. You can also find a bottle during the mission 'Tunnel Rat', on some boxes by a door that leads to your target. You must sneak up to your target of choice with the Anesthetic equipped and press to use it. You will know it has been successful, as you will smother your target with a cloth and they will be taken to the floor unconscious.
You must be behind your target and in sneak mode, or else you cannot use the Anesthetic.