Is that one of those quests where you need to capture an enemy after getting it's HP down to critical? I've done countless levequests like that, as those are really tricky when you're at a high level, but they are still possible, you just can't go in with all guns blazing as the enemy is quickly killed as you noted.

The way I do it is by calling out my chocobo and letting them attack while not attacking myself, then when the enemy is almost dead, immediately commanding the chocobo to cease attacking, and then using the item to capture the enemy.

It's tricky, but definetely doable (you might have to fail it a few times to get the hang of the strategy).