She won what no woman had won before in Iowa.

Hillary Clinton was declared the winner of the Iowa caucuses on Tuesday after final vote counts showed her narrowly beating Senator Bernie Sanders of Vermont, according to The Associated Press and other news organizations.

“I can tell you, I’ve won and I’ve lost there — and it’s a lot better to win,” Mrs. Clinton said to cheers at Nashua Community College.

According to the final results announced by the Iowa Democratic Party, Mrs. Clinton was awarded 700.59 state delegate equivalents, the terminology used in Iowa to represent candidates’ share of the total caucus vote. Mr. Sanders was awarded 696.82 delegates, and former Gov. Martin O’Malley received 7.61 delegates. Iowa Democrats usually do not release raw vote counts from each of the state’s 1,681 caucus precincts.

The Iowa results — the closest in the history of the Iowa Democratic caucuses — were far tighter than Mrs. Clinton and her advisers had expected. But her team embraced the result as a significant win given her previous troubles in the state — she came in third place in the 2008 caucuses — and Mr. Sanders’s popularity with many liberals here. (In a poll last month, 43 percent of likely Iowa Democratic caucusgoers described themselves as socialists.) Still, the close margin privately frustrated Mrs. Clinton and former President Bill Clinton, according to advisers, because they had hoped that a strong win would give her much-needed political momentum in New Hampshire, where Mr. Sanders has a solid lead in opinion polls.