What is the state in which a person is put to sleep using anaesthesia called?

What you are looking for is

I was under general anaesthetic during the operation.

‘Anaesthetised’ is not explicit about the anaesthesia being general rather than local, although that would be the usual assumption. With local, one might say ‘I was numbed up’ or ‘they numbed my arm’ or ‘they anaesthetised my leg’ or ‘they gave me an epidural’.

‘Sedated’ is wrong if I’ve understood what you’re asking about. Sedation is less extreme than general anaesthetic. Sedation is induced sleep, whereas general anaesthesia shuts your brain down on a deeper level. You can usually be woken from sedation by shaking or shouting (it depends on how much they give you), although you will fall straight back to sleep and forget the interaction. With sedation, you need local anaesthetic too, or else the pain of cutting is likely to wake you. Milder forms of sedation (e.g. taking a benzodiazepine for anxiety) won’t even put you to sleep.

The straight answer would be anesthetized/anaesthetized/anaesthetised because it is the verb form of anaesthesia (the very word you used in your question). It is acknowledged that another answer already mentions "under anaesthesia" (which is not bad but is not a single word to fit the blank in the example sentence).

I was anesthetized/anaesthetized/anaesthetised during the operation

MW (medical dictionary):

anesthetize
transitive verb
variants: or chiefly British anaesthetize also anaesthetise

anesthetized (or chiefly British anesthetized also anesthetised; anesthetizing or chiefly British anesthetizing also anesthetising) :
to subject to anesthesia

Yet another (probably more common) word is sedated.

I was sedated during the operation

MW:

sedate
transitive verb
: to dose with sedatives

The doctor sedated the patient heavily.

Another one: drugged to mean (rendered) unconscious due to drug(s).

I was drugged during the operation

M-W:

drug
drugged; drugging
transitive verb
1 : to affect with a drug (see drug); especially : to stupefy by a narcotic drug

looks like he's been drugged

Usage examples of drugged which match the example sentence: