Iran is prepared to engage in "time bound talks ... with full transparency," Rouhani said, adding that Iran "does not seek to increase tensions with the United States."
Rouhani said Washington must speak with one voice and avoid "war mongering."
The Iranian leader spoke only hours after U.S. President Barack Obama addressed the General Assembly. Obama said Iran's positive statements must be followed by positive actions that can be verified.
Rouhani said Iran's nuclear policy calls for its nuclear program to "pursue peaceful purposes."
"This has been and will always be Iranian policy," he said. "Our national interests make it imperative that we remove any and all concerns about Iran's nuclear program."
He said the world must "respect the right of Iran" to pursue nuclear energy.
Western governments and Israel have maintained Iran is close to developing a nuclear weapon. Obama has refused to take the military option off the table if that event gets near.
Rouhani repeatedly condemned Western economic sanctions, which he said are causing suffering in his country.
"These sanctions are violent, pure and simple," Rouhani said. "These sanctions violate inalienable human rights ... sanctions beyond any and all rhetoric" causing "war mongering and human suffering."
Rouhani's comments at the United Nations were far less controversial than those of his predecessor, Mahmoud Ahmadinejad, who led Iran from 2005 to 2013. He denied the Holocaust and harangued the delegates on the evil of the United States during his U.N. speeches.
Rouhani shared the Ayatollah Ruhollah Khomeini's exile in Paris in the 1970s. Time magazine reported he speaks fluent Farsi, English and Arabic, and has a doctorate from a Scottish university.
Far from denying the Holocaust, Rouhani quoted the Koran and the Torah to say the meek would inherit the Earth.
"I am deeply optimistic about the future," he told the General Assembly. "My hope ... emanates from the belief shared by all divine religions, that a good and bright future awaits the world."