Rouhani also addressed Iran's role in the Syrian civil war.
Some Iran troops are fighting on the side of President Bashar al-Assad's regime, Amanpour noted.
Rouhani insisted the Iran's involvement does not even number hundreds of people.
"We have close relations with Syria from a long time ago," he said, adding that Iran has military attaches and experts stationed there.
Asked about Iranian weaponry being used by the Syrian regime, Rouhani cautioned against baseless "propaganda."
"Are you encouraging the regime to give up its chemical weapons as the deal between the U.S. and Russia says?" Amanpour asked.
"We believe in general that the entire region of the Middle East has -- as far as that region is concerned -- all weapons of mass destruction, including chemical weapons, have to be eradicated from the region," Rouhani responded.
Iran is glad Syria has committed itself to the Chemical Weapons Convention, he said.
Rouhani on why no meeting with Obama
"I would like to say to American people: I bring peace and friendship from Iranians to Americans," Rouhani said in English, marking the first time he's spoken the language in a TV interview since becoming president. For the rest of the interview Rouhani spoke Farsi.
In many ways, Rouhani is the "it" man of the U.N. General Assembly, as Western leaders look to gauge whether his diplomatic overtures will translate into concrete policy changes.
There was widespread speculation that he and U.S. President Barack Obama might meet face-to-face.
"There were some talks about" a possible meeting, Rouhani told Amanpour through a translator. "And preparation for the work was done a bit as well."
But no such meeting happened.
Two senior U.S. administration officials told CNN Tuesday that the encounter was called off because it was considered "too complicated" for Iran back home.
"I believe we didn't have sufficient time to really coordinate the meeting to the full extent that we needed to," Rouhani told CNN.
Given that U.S. and Iranian leaders have not met face-to-face for 35 years, he said, "we must give time for diplomacy to work itself, for dialogue to come about, for circumstances to be laid properly."
Still, the two men recently exchanged letters, and Rouhani said the ice is "already beginning to break because the environment is changing and that has come about as a result of the will of the people of Iran to create a new era of relations between the people of Iran and the rest of the world."
Resolving the standoff over Iran's nuclear program is the priority, he said. "If the nuclear issue is settled conclusively, I believe that that will pave the way for numerous other issues that can be discussed."
Iran insists its program is for energy purposes; the United States and several other countries believe it could be a guise for building nuclear weaponry.
Rouhani also called for sanctions against his country to be lifted.
Rouhani said he has full permission from Iran's supreme leader, Ayatollah Ali Khamenei, to negotiate with the West.
"I think that the president of Iran has the authority whenever the national interest of the country is involved," Rouhani told Amanpour. "The supreme leader of Iran has said that should negotiations be necessary for the national interest of the country, he is in fact not opposed to it."
"Now, if an opportunity was created today, had arisen today," the Iranian president said, "and the prep work for that had been done, most probably the talks would have haven taken place, primarily focused on the nuclear issue or the developments on the Middle East. Therefore the supreme leader, I can tell you, has given permission for my government to freely negotiate on these issues."
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