NEW YORK (CBSNewYork/AP) — Hillary Clinton handily won the New York Democratic presidential primary Tuesday night.

With 55 percent of the precincts reporting, Clinton had 59 percent to 41 percent lead over rival Bernie Sanders.

CBS News polling showed Clinton won 57 percent with women voters to 43 percent for Sanders, 70 percent with older voters compared with 30 percent for Sanders, and 71 of black voters compared with 27 percent for Sanders.

Eighty-nine percent of exit poll respondents said Clinton had the right experience, compared with only 11 percent for Sanders.

PHOTOS: Hillary Clinton Holds Rally After New York Primary Victory

In addressing a cheering crowd at the Sheraton New York near Times Square, Clinton said her victory in New York – where she has lived for 15 years and served in the U.S. Senate – is “personal,” CBS2’s Tony Aiello reported.

“New Yorkers, you’ve always had my back, and I’ve always had yours,” Clinton said. “Today, together we did it again, and I am deeply, deeply grateful.”

Clinton also reached out to Sanders supporters.

“To all the people who supported Senator Sanders, I believe that there is a lot more that unites us than divides us,” she said.

But without naming Sanders, Clinton did touch on a frequent criticism of her Democratic presidential rival that his policy plans are not practical.

“It is not enough to diagnose problems. You actually have to explain how you solve the problems,” she said.

She drew cheers as she called for action on climate change, comprehensive immigration reform, a guarantee of equal pay for women, and a commitment to protect the rights of all. While not mentioning Republican candidate Ted Cruz at that point, she did reference his disparaging remark about “New York values” as decadent and liberal.

“Civil rights, voting rights, women’s rights, workers’ rights, LGBT rights, and rights for people with disabilities — those are after all New York values. They are American values,” she said.

Clinton later did name Cruz and GOP frontrunner Donald Trump by name, and took them to task for what she said was a call for a “return to trickle-down economics,” and efforts to round up immigrants, and keep Muslims out of the country while treating American Muslims like “criminals.”

“These things go against everything America stands for,” she said.

Before Clinton took the stage, former Mayor David Dinkins and Mayor Bill de Blasio were among the speakers. De Blasio led a call and response cheer with the crowd, where he said, “If this nation is going to fight back the forces the hatred and division,” and, “If this nation is going to be all it was meant to be,” with the crowd responding, “We need Hillary!”

Gov. Andrew Cuomo came next, likewise leading the crowd in thunderous cheers.

“Hillary is a New York Democrat. She gets us!” Cuomo said. “She is a can-do Democrat. She knows that government is about making a difference in people’s lives and getting results. She knows in New York state, we don’t just talk the progressive talk. We walk the progressive walk. She knows in New York state, we deliver results.”

But the crowd went silent when Erica Lafferty, the mother of Sandy Hook Elementary School Dawn Hochsprung was in the room. She said Lafferty had advocated for common-sense gun control, and a world where people were in danger of gun violence cannot be tolerated.

“That’s not the type of world I want to live in, and we refuse to live in that,” she said.

Clinton ended by invoking the state’s Latin motto.

“The motto of this state is excelsior, ever upward. So let’s go out and win this election and all rise together,” she said.

Clinton’s campaign had cast New York as a make-or-break moment for the Democratic race. A loss in her adopted home state would have been devastating political blow. But a victory will bolster her delegate lead over Sanders and put her closer to becoming the first woman nominated for president by a major political party.

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Sanders, a Brooklyn native, had found a warm welcome of his own in New York before the primary. He turned out impressive crowds at enthusiastic rallies – including one in Long Island City, Queens, Monday night.

“This is a campaign on the move,” Sanders shouted to a crowd of thousands gathered along the waterfront in Queens. “This is a movement getting the establishment very, very nervous.”

But late Tuesday, Sanders had already left New York and his campaign announced that he would not be speaking.

Sanders appeared on the campus of Penn State University in State College, Pennsylvania, with an eye on the next round of primaries on April 26. He told supporters he thinks Clinton is nervous that her lead in the delegate count is slowly being eaten away.

Sanders also complained that New York has a closed primary, in which voters must register as a Democrat or Republican in October to vote in the April primary.

As of 11 p.m., Clinton had captured 129 new New York delegates, while Sanders had 98. Twenty more remained unaccounted for.

Before the New York primary, Clinton had accumulated 1,755 delegates to Sanders’ 1,060, according to CBS News. Those totals included both pledged delegates from primaries and caucuses, as well as superdelegates, the party insiders who can back the candidate of their choice regardless of how their state votes. It takes 2,383 to win the Democratic nomination.

But while the race has been contentious, a poll showed that 68 percent of New York Democratic voters had been energized, rather than divided, by the contest.

(TM and © Copyright 2016 CBS Radio Inc. and its relevant subsidiaries. CBS RADIO and EYE Logo TM and Copyright 2016 CBS Broadcasting Inc. Used under license. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.)