Ban will succeed Kofi Annan, who has headed the world body since 1997. The new UN chief will assume the post on January 1.
The appointment was based on a recommendation of the UN Security Council.
Ban told the General Assembly that he was proud to become the second Asian ever chosen to head the UN, following U Thant of Burma, who served from 1961-1971.
Ban promised "to work diligently to materialize our responsibility to protect the most vulnerable members of humanity and for the peaceful resolution of threats to international security and regional stability."
Ban said fighting poverty, HIV/AIDS, and environmental degradation would be some of his priorities as UN chief. He also said protecting human rights, containing the spread of weapons of mass destruction, and combating terrorism were high on his list.
In his speech, Ban referred to his reputation as an unassuming politician, saying that "Asia is a region where modesty is a virtue," but, he continued, "modesty is about demeanor, not about vision and goals. It does not mean the lack of commitment or leadership. Rather it is a quiet determination in action to get things done without so much fanfare."
Ban will take the helm of the world body at a particularly tumultuous time. For the past few years, the organization has been plagued by accusations of mismanagement, inefficiency, and sex abuse by its peacekeeping units. The Iraqi "oil-for-food" corruption scandal, involving billions of dollars of misappropriated money, severely tarnished the UN's image and raised doubts about the capability of the organization to fulfill the ideals declared in its founding charter 61 years ago.
The 62-year-old diplomat pledged to push forward with large-scale reforms of the UN, the most far-reaching in the organization's history, that were launched by outgoing chief Annan in response to criticisms of corruption and mismanagement.
"We must reform because we believe in the future," Ban said. "To revitalize our common endeavor is to renew our faith not only in the UN's programs and purposes but also in each other.'
Ban is well-acquainted with the UN. He has worked within the organization in various capacities for more than 10 years. His selection is also considered to have been influenced by the UN's so-called "geographical principle" -- that is, the notion that the post should be rotated between continents.