This is a very late answer and doesn't really substantially differ from the previous answer, nevertheless, I thought I could at least add a few pointers to places in Leibniz's texts.
Roughly speaking you are correct. Leibniz, to my knowledge, never states that this is the best of all possible worlds, but it is the conclusion that Voltaire---not without reason---thinks he must reach.
Leibniz tends to argue much as you phrase it from what he takes to be the fact of God's perfection. His argument, however, is not that God could not have acted less perfectly in creating this world, but that there is no reason why God should have acted less perfectly than he was able to in creating this world. (If God had acted less then perfectly, then there would be a sufficient reason for him doing so and not otherwise). (Something like this argument appears in both his correspondence with Clarke and his text known as the "Discourse on Metaphysics").
But, of course, as you state, there is obviously evil in the world, much of which seems unnecessary. For Leibniz, however, we only have a limited perspective on things and what seems unnecessary to us may be necessary precisely because it is the only thing "compossible" with the maximal amount of goodness. In other words, God didn't create a world free from evil because there is a reason that such a world would not be compossible with the maximal amount of goodness, and hence God would've acted less perfectly than he could've.
Compossibility actually gets us much closer to why Leibniz talks about "possible worlds"---it has little to do with a need for comparison. One of Leibniz's earliest mentions of this comes early on in his correspondence with Arnauld where he is discussing the creation of Adam. Leibniz argues (in his "Discourse on Metaphysics" which prompted the correspondence) that having conceived of Adam, God, being omniscient, necessarily conceived of all that would happen from creating Adam, i.e., the entirety of human history. Arnauld objected that this would seem to make God responsible for all the evil that came into the world starting with Adam. Leibniz's response is along the lines of yes, but even knowing the evil that the creation of this Adam (out of all possible Adam's) would bring, God still chose to create this Adam. Since God has no reason to act less perfectly than he is capable of, he must have chosen the best of all possible Adams, which is to say the Adam whose history is compossible with the maximal possible goodness.
This is where Voltaire who, importantly, lived after 1755, saw a problem with Leibniz's arguments. Since, above all else, the earthquake in Lisbon seemed to have caused more damage than could possible justified by any amount of future goodness. (And recall the quake is central to the horrors of Candide). We can only speculate how Leibniz might've responded.
Nevertheless, Leibniz, quite famously, is never happy with just one way of arguing for things, and often will in one place present arguments for what in other places he takes as axioms and so on. Leibniz's arguments for God having created the world in the most perfect way are thus, for him, compossible with a range of other arguments. All of which is to say that any attempt to give "Leibniz's knock-down argument" is doomed to failure from the start.
I'll make one last observation, however, which is that Leibniz never understood himself to be engaging in theology. The problem of evil, as he addresses it, is a real problem, but a philosophical, rather than theological problem. (It's worth pointing out that Leibniz resuscitated a version of the ontological argument and so took God's perfection to be philosophically demonstrable). At one point in his correspondence with Clarke, he mocks Clarke for relying on miracles to explain the natural order which, according to Leibniz, philosophy always tries to avoid.