Given that WLC's burden of proof was to show that God's existence is more probable in light of contemporary cosmology that it would have been without such cosmological data, WLC would have had the lighter burden of proof; indeed, a much, much lighter burden than Carroll's. My assessment of the debate isn't so much based on what WLC did - since I'm familiar with his material and just needed to see how Carroll would respond - but rather my assessment is based on how Carroll responded. While I'm not working with a transcript or playing through the video as Carroll goes through his points, I'll try and give a general overview of Carroll's response.
Carroll's first main point was that naturalism "works" in the sense that it accounts for the data we observe. In other words, naturalism is a better scientific hypothesis than theism. Carroll gives some examples as to what we should observe if theism were true, and I didn't find any of them to be particularly convincing. For example, Carroll says if theism were true, he'd expect God's existence to be much more obvious. This has been addressed by WLC, so I won't just repeat a response that's already been given. All of the points he raised were either already addressed - as anyone with a slight familiarity with WLC's material knows - or were simply presumptuous. Carroll didn't explain why he is in such a privileged position as to know what God would or wouldn't do if He existed.
One of the issues Carroll kept mentioning was his claim that theism is not well-defined. Obviously, theism is pretty easy to define - "The belief that God exists" - so that's not really what Carroll means, I would think. It seems to me that Carroll is suggesting that God is what's not well-defined, yet this objection only holds if God is a scientific hypothesis. This is another claim Carroll likes to bring up. He seriously seems to think that God is just a hypothesis, and He should be subjected to the same rigorous analysis any other scientific theory would have to endure. Yet I don't think any sophisticated theist - least of all WLC in this debate - posits God as just an explanation of our cosmological observations and data. Why Carroll then treats God as if He is just a hypothesis is beyond me. It seems like it would be more productive to critique your opponent's actual view.
Moreover, Carroll fails to make a basic distinction between How things are and How things came to be. Yes, naturalism can provide a plausible account of how reality is. (In fact, apart from the naturalist assumption that the universe is a closed system, the theistic account is almost identical to the naturalist one.) But that's not the question that's raised in the kalam, for example. Carroll repeatedly says we have no right to demand anything more than a model that is self-contained and fits the data. But the obvious presupposition of such a statement is that there is such a model. Besides the Hartle-Hawking model, he really doesn't say which model is both eternal and supported by empirical data. Really, Carroll kept making an implicit statement of faith that future cosmological models based on a better understanding of quantum theory will fill the post of being both eternal and based on the evidence. But the debate was really about modern cosmology.
Carroll implicitly concedes the point to WLC. Modern cosmology does support the notion the universe had a beginning, but future cosmology will sort all that out. In essence, Carroll concedes premise 2 of the kalam fits nicely with current cosmology.
As far as the first premise of the kalam goes, Carroll kept bringing up possible models that come into being uncaused or never began (thereby shifting his response to premise 2). Indeed, Carroll never really responded as to why he thought the universe could come uncaused from nothing; he simply directed us to eternally existing models. To deny that premise 1 is more probable than not - which is all we're looking for in a good premise - seems unconscionable for a sincere, open-minded seeker of truth. Regardless of a person's personal psyche, what they assume in denying premise 1 is that the universe could come into being - uncaused - out of nothing. Yet such an assumption (1) violates a metaphysical principle that pre-dates Socrates and (2), as Carroll would say, "doesn't meet our expectations." If the entire universe - defined as all physical reality - can just come into being from non-being, why is it that anything and everything doesn't follow suit? It seems to me such an expectation is pretty natural, given that we're meant to entertain the idea a whole universe could come from literally nothing.
Again, Carroll's response was just to tell us there are models that circumvent that problem. No doubt that's true, but those models aren't themselves perfect, as WLC pointed out. So Carroll really didn't give us any reason to doubt premise 1 is more probable than not. Indeed, I don't think any rational and sincere person can deny it with a straight face (unless they want to pull a Dennett and say the universe caused itself).
Obviously this analysis was just about some of Carroll's general remarks and his response to the kalam, but I think it shows that Carroll did not refute premise 1, implicitly admitted premise 2 fits with modern cosmology, and failed to substantiate his presumptuous expectations about reality if God exists.