First let me give you a disclaimer. This is an analysis made with what would be considered poor data, so to what extent the debate went in either direction is still up for debate. However, I feel confident in at least declaring the winner with some certainty, but things will become more clear as time goes on and more information is gathered.

If you just want to know my answer without reading about what I base it on, you will find it at the bottom. Starting off, I want to first mention which sources I base this analysis on and more importantly WHY I chose them instead of all the others.
First we have the early scientific poll done by CNN and ORC. What I would like to say is that "scientific" means it is reliable but sadly, all too often it's just an empty word thrown around which makes it harder to explain why this can be trusted.
The poll itself can be found here:
https://i2.cdn.turner.com/cnn/2016/images/09/27/poll.pdf
The poll included 521 people randomly picked nationwide:
26% Identified themselves as Republicans
41% Identified themselves as Democrats
33% Identified themselves as Independent
In terms of makeup it's not an ideal spread since the national average is slightly different but it is made up by the clarity of the poll results which overcomes this disparity.
The reason I can trust this poll is because it formats the questions as best it can to not influence the response, and they also contacted them through phone which is a more personal form then holding an internet survey, where the people asked are picked randomly instead of letting anyone submit their vote. While it is not ideal for a definite answer it is reliable enough to make broad statements such as if it was close or if one side won comfortably.
We also have a larger poll which is very similar to the CNN/ORC one made by Public Policy Polling.
https://www.publicpolicypolling.com/pdf/2015/PPP_Release_PostDebatePoll_92616.pdf
The poll included 1002 people:
32% Identified as Republican
41% Identified as Democrat
27% Identified as Independent
My biggest concern about this poll is that I am not entirely sure how they selected their sample but they have had good results previously and can be trusted to not make such a massive breach of polling etiquette by deliberately choosing who to ask to skew the results.
I will reveal the poll results and my analysis of it further below along with the rest of the results.
Secondly we have the focus groups. What focus groups are is a collection of undecided voters that are put together to watch it and give their opinions. By the nature of this its very different from polls since its a measurement of a subset that may or may not be biased. They are also usually very small which does not help the matter.
As such I will not be relying on them for the final analysis of who won but they still provide important insight in peoples reaction to the specific questions in the debate and so on. As such I will mostly rely on them on what both candidates did right and wrong in the debate.

=============================
Poll results and analysis
The results of the two major polls from CNN/ORC and PPP both agreed that Clinton performed best in the debate and also won more new support.
CNN/ORC poll on best debate performance
62% - Clinton
27% - Trump
PPP poll on best debate performance
51% - Clinton
40% - Trump
Looking at these two polls it appears that Clinton won the debate fairly clearly. While I can't tell you by how much from these early and small polls, it's a significant enough margin and it's safe to say Clinton had a comfortable win.
CNN/ORC poll about the effect on their support after the debate.
47% - No effect
34% - More likely to vote for Clinton
18% - More likely to vote for Trump
PPP poll about being more or less likely to vote for Clinton after the debate.
40% - More likely
35% - Less likely
23% - No difference
PPP poll about being more or less likely to vote for Trump after the debate.
35% - More likely
39% - Less likely
24% - No difference
What to take note of in these results is that Clinton is leading comfortably in new/stronger support, but also notice that Trump, while gaining around as much as Clinton, actually also had a very high negative as well, which even surpassed the positive. Not looking too closely at the numbers because they are unlikely to be too accurate, it appears that Trump has problems with polarization. While he grows his support he also grows his opposition.
Looking at the focus groups to try find out what resonated with the viewers as well as what mistakes they made, it appears that at least from the start Trump had a good impression among viewers. However, throughout the middle part and latter part Clinton recovered. This supports Trump's character of going in hard and strong with crude attacks against Hillary but lacking in substance which becomes more apparent the longer the debate continues.
According to Fox News focus group (I know its not the best of sources normally but we are not here for accurate statistics but to know the reaction of the people in the focus group) the topics that hurt Trump the most was regarding Birtherism and to a lesser extent questions about taxes.
Knowing this, it's likely that Trump will push for shorter debates in the future.

TL:DR
The current polls indicate that Hillary won but its unclear by how much. Trump is having problems appealing to people without getting a negative reaction and he burns himself out on debates if they go too long.
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1Opinion
Good and sound analysis but they were all in hours after the debate but the true figures (polls wise) won't come through the weekend to see if any change in support for candidate. I predict the Donald Trump vote won't change, the question is how much does Hillary Clinton's vote change. I would say pre debate HC was ahead by approx 1.5 in national polls and tied in battleground states. I think HC will be 4/5 points ahead in new national polls and maybe get a 1/2 points bump in the battleground states.
Yeah it will be interesting to see how things change from here because people really expected more from Trump then what he gave to them.
6Good post