The UN appearance was Malala's first public speaking engagement since a Taliban gunman shot her in the head last October in a bid to end her campaign to get girls into schools.
"The attack on October 9, 2012 was just a part of my life," Malala said at a reception at the Pakistani UN mission in New York.
"I want to work hard, I want to sacrifice my whole life for the education of girls.
"And to be true, I want to say that I don't to be the girl who was shot by the Taliban, I want to be the girl who struggled for her rights."
The teenager, considered a strong candidate for the Nobel Peace Prize, said she was determined to keep her her struggle "for a right to live in peace, for a right to go to school."
But she reaffirmed her message that the Taliban and other extremists "do not understand the importance of education."
The Taliban were among "people who think that when a woman goes to school she will be empowered, and they are afraid of it," she emphasized.
"They are still targeting schools, they are still killing innocent children," she said, referring to recent attacks both in her native Pakistan and Nigeria.
"If we work together, we will soon see that there will be many schools created in Pakistan and Afghanistan and poor countries. And we will see that every woman and every girl will have the same rights as men have," she said.
"We do want equality, we are not like men," she joked.
Malala is expected to return to New York for a summit on education on the sidelines of the UN General Assembly of world leaders in late September.
"Malala's speech was just the start of a momentous push for change in the run up to 2015, to deal with the education emergency," said Gordon Brown, the former British prime minister and now UN special envoy on global education.
Getting all children into primary school by 2015 was one of the Millennium Development Goals agreed at a world summit in 2000.
Malala was given several standing ovations for her speech Friday when she said she would not be silenced by the Taliban.
She declared: "Let us pick up our books and pens. They are our most powerful weapons. One child, one teacher, one pen and one book can change the world."
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