Wonderful word, this one. Common usage is as an alternative to "great" that sounds more expressive.
1. Executed with exquisite skill.
Coupling 'executed' with 'exquisite' and the further x of 'skill', and the whole phrase is redolent of finesse, precision and expertise.
2. The exquisite pain of romance.
Here the usage is possibly just the same - 'exquisite' is being used as an emphasizing word that has neutral connotations. Compare it with the following;
3. The wonderful pain of romance.
4. The terrible pain of romance.
'Wonderful' and 'terrible' are both providing emphasis, but the former with a positive slant, the latter with a negative. It is conceivable that the writer of 2 was aware of the precise denotation - even the derivation - of the word, but you can see how it may be used as an alternative to 'consummate' (adj.) which just means "complete" or "full" - or as a truss-word - it strengthens the structure of meaning without adding details.
One of the consequences of the use of exquisite purely for emphasis is that the adverb exquisitely often ends up as a fancy alternative to very. This, in my opinion, is a pity.
Consider the "standard" definition of exquisite which is "fine, precise, detailed, delicate", the which arises from the etymology, which means "sought-after". Here's why I think it's a wonderful word. If you didn't know the etymology, you are probably still using the word correctly. Consider 1 in the light of this; 'exquisite skill' would likely increase the value of whatever is being done so skilfully, hence make it more sought-after. Consider 2, and romance is certainly something that continues to be sought, even once we know that it can be painful.
The reader who appreciates the meaning of exquisite will feel that it has added meaning in these cases, whereas the reader who does not will not be further enlightened by your usage, but will nonetheless get a sense of what you seek to say.
This sense of rarity gives exquisite its connotation of desirability, and explains why the bad guy often uses it to describe unpleasant things. It's because he's educated, probably British, and he knows what it really means, but he's messing with the hero's head because he knows that the hero is blue-collar and thinks that exquisite means expensive.
5. You will be tortured exquisitely until you talk.