(CNN) -- Solid support from registered Democrats and women in New Hampshire were crucial Tuesday as Sen. Hillary Clinton rebounded from her third-place finish in last week's Iowa caucuses.

Sen. Hillary Clinton has spent the past few days saying she has the experience to change Washington.

She narrowly defeated Sen. Barack Obama in the New Hampshire primary, with 39 percent of the vote to Obama's 37.

"Last week, I listened to you, and in the process I found my own voice," the New York senator said after her victory.

"Now let's give America the kind of comeback that New Hampshire has just given me."

Forty-three percent of self-styled independents said they voted for Obama, and 31 percent said they backed Clinton. Independents made up 43 percent of all voters polled.

Addressing his roaring supporters after the race was called, Obama congratulated Clinton. But he was a candidate determined to draw a distinction between Clinton and himself.

"But the reason our campaign has always been different, the reason we began this improbable journey almost a year ago, is because it's not just about what I will do as president," he said. "It is also about what you, the people who love this country, the citizens of the United States of America, can do to change it. That's what this election is all about."

But Clinton was ahead of Obama 45 percent to 34 percent among those who said they were registered Democrats. Those voters made up a majority -- 54 percent -- of all respondents.

Clinton also claimed the majority of women's votes, according to the polling. That's in contrast to last week's Iowa caucuses, in which Obama surprised observers by stealing the female vote from Clinton.

Analysts say that shift among female voters was crucial to the Clinton turnaround. "If I had a single word, the word would be 'women,' " said CNN political analyst Bill Schneider. "She got the women back."

And Schneider said the support of union voters that put Clinton over the top. "Union voters have her a 10 point lead," he said.

CNN projected former Sen. John Edwards to finish third.

College graduates, who made up 29 percent of the electorate, opted narrowly for Clinton -- 38 percent to Obama's 37 percent, according to the polling.

Those polled who called themselves very liberal, and moderate, went with Clinton over Obama -- although by less than 2 percentage points in each -- and those who said they are somewhat liberal were evenly split.

Pundits also were citing the role of former President Bill Clinton in helping his wife recover from what pre-primary polls were suggesting was a deficit of 9 percentage points to Obama in New Hampshire.

The former president spent Tuesday in Hanover -- home to Dartmouth College -- where Obama had been expected to win handily.

"They dispatched him to the area that Obama was surging," said CNN analyst Donna Brazille, who managed former Vice President Al Gore's campaign in 2000. "I think it had the effect of tamping down Obama support and giving Senator Clinton a real reason to come back in this race."

New Hampshire was considered crucial to Clinton's campaign. If Obama had been able to sweep Iowa and New Hampshire -- after months of Clinton being considered the front-runner among Democrats -- it could have given him powerful momentum going into future primaries.

"Age is also playing a big factor -- older voters are overwhelmingly outnumbering younger voters -- a proportion that is clearly benefiting Clinton," Schneider said. "Sixty-seven percent of Democratic primary voters are over the age of 40, and they are breaking heavily for Clinton over Obama."

Over the past several days, Clinton has trumpeted her experience, saying that she has delivered change as both first lady and as a senator.

After losing to Obama in last week's Iowa caucuses, it was unclear whether she could overcome what appeared to be Obama's ability to electrify American voters who had previously taken a sour and skeptical view of politicians and the political process.

The duel between the Obama and Clinton campaigns grew especially testy Monday and Tuesday. She said she had more experience than he, and was therefore more qualified. He accused her of representing the status quo of Washington.

And on the eve of the New Hampshire primary, Bill Clinton criticized the media for not pressing Obama more fully on Iraq, and accused the Illinois senator of shifting his position to reflect changing attitudes about the war in Iraq.

Then, there was an issue unto itself -- Hillary Clinton's almost-tears.

Clinton's eyes welled up this week while responding to a voter's question about her health and appearance.

Pundits and voters alike questioned whether Clinton's emotions were sincere or faked as part of some strategy to diminish criticism that she is too steely, too cold.

In front of the crowd of mostly female New Hampshire voters, an admittedly fatigued Clinton responded to a question by saying: "This is very personal for me, it's not just political, it's [that] I see what's happening, we have to reverse it."

Her voice broke, and she was then applauded by the crowd.

Polls indicated the show of emotions fared well with male voters, according to CNN's John King, but turned off some female voters.

Edwards was followed in votes by New Mexico Gov. Bill Richardson and Ohio Rep. Dennis Kucinich. E-mail to a friend

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