Aides to President Bush had said in advance the speech would focus on issues the world community can, and should, agree on.
He called on the General Assembly to rally behind the 1948 Human Rights Declaration that commits U.N. members to liberate their people from tyranny, violence, poverty and disease.
Mr. Bush said the best way to overcome terrorism and extremists is to defeat what he termed "their dark ideology" with a more hopeful vision.
He saluted countries that have recently taken strides toward liberty, including Ukraine, Georgia, Kyrgyzstan, Mauritania, Sierra Leone and Morocco. He said every civilized nation has a responsibility to stand up for people still suffering under dictatorship in countries like Belarus, North Korea, Syria, and Iran.
But he put special stress on Burma, the scene in recent days of the largest pro-democracy demonstrations in many years against the country's military authorities. He said Americans are outraged by what he termed a "19-year reign of fear" in Burma and the continued detention of democracy leader Aung San Suu Kyi and other political prisoners.
President Bush said his administration will increase sanctions against the Rangoon government to try to bring about peaceful change.
"The United States will tighten economic sanctions on the leaders of the regime and their financial backers," he said. "We will impose an expanded visa ban on those responsible for the most egregious violations of human rights, as well as their family members. We will continue to support the efforts of humanitarian groups working to alleviate suffering in Burma, and I urge the United Nations and all nations to use their diplomatic and economic leverage to help the Burmese people reclaim their freedom."
In remarks that prompted a walkout by Cuban delegates, Mr. Bush said the long rule of a cruel dictator is nearing its end in Cuba, and said as that country enters a period of transition, the United Nations must insist on free speech, assembly and, ultimately, free elections.
He said in Zimbabwe, the behavior of what he termed the tyrannical regime of President Robert Mugabe is an assault on the country's people, and an affront to the Universal Declaration. He said the United Nations must insist on change in Zimbabwe, and answer the challenge to conscience posed by the Darfur crisis in Sudan by prompting deploying an expanded peacekeeping force there.
The president said the founding goals of the United Nations, including liberating people from hunger and disease and illiteracy, cannot be achieved without reform of the organization. He said his administration is committed to a strong and vibrant United Nations, but is troubled by the outcome of the nominal reform of the world body's human-rights apparatus.
"The American people are disappointed by the failures of the Human Rights Council," he said. "This body has been silent on repression by regimes from Havana and Caracas to Pyongyang and Tehran - while focusing its criticism excessively on Israel. To be credible on human rights in the world, the United Nations must first reform its own Human Rights Council."
Mr. Bush urged the international community to support moderate leaders in the Palestinian territories and to stand with he termed the young democracies of Lebanon, Afghanistan, and Iraq.
The president went directly from the General Assembly speech to a bilateral meeting with Iraqi Prime Minister Nouri al-Maliki and is due to meet with Afghan President Hamid Karzai before wrapping up three days of U.N. diplomacy on Wednesday.