When legions of DC insiders concluded that Hillary Clinton won the Democratic debate on Tuesday, I was instinctually more skeptical. Insiders had badly misjudged what actual voters thought of some recent debates, and there was a case, I thought, that Sanders — who put in a solid performance and who is still relatively little-known nationally — could have picked up some support.

But now that the post-debate polls have started coming in, we're getting evidence suggesting the insiders got this one right.

NBC News/SurveyMonkey, the Huffington Post/YouGov, and Gravis/One America News have all released polls finding that Clinton won the debate by huge margins among Democratic voters who watched it:

Now, the first two are online polls (though conducted with much more methodologically rigorous techniques than the simple online surveys showing Sanders won), and the third is an overnight robocall poll from a minor firm. But they all ended up with very similar results, and the NBC/SurveyMonkey poll in particular is highly respected in the field as using sound methodological techniques.

It's less clear, however, that Clinton's apparent win actually boosted her level of support. NBC/SurveyMonkey found that she picked up 3 percentage points and Sanders picked up 2 (with Biden being the most hurt). HuffPost/YouGov only asked about whether Clinton or "someone else" should be the nominee, and Clinton picked up 8 points in that question.

And the dust hasn't entirely settled yet. Another thing to look at is how Sanders's numbers move in the next few days. The Vermont senator has been stalled out at about 25 percent support in national polls of Democrats since early September. So it's possible that even if he "lost" the debate in the view of most Democrats, he could have won some new supporters and boosted his numbers higher.

But the evidence so far suggests that Clinton has a lot to be happy about — and that the instincts of DC insiders were generally on target.

Update: A new poll of New Hampshire likely Democratic voters, from Suffolk University and the Boston Globe, was also released Friday, with very similar results. 54 percent thought Clinton won the debate, compared to 24 percent for Sanders. Even worse for Sanders, the poll showed Clinton reclaiming a narrow lead in New Hampshire (she's trailed Sanders in every poll of the state since early August).

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