Speaking after Kerry, Zarif said the meetings had been "very constructive" and "very businesslike."
"We hope to be able to make progress to solve this issue in a timely fashion (and) to make sure (there is) no concern that Iran's program is anything but peaceful," he said.
"I am satisfied with this first step," Zarif said. "Now we have to see whether we can match our positive words with serious deeds so we can move forward."
He said the end result would have to include "a total lifting" of the international sanctions that have devastated Iran's economy.
Ashton called the meeting "substantial" and announced that the parties had agreed to "go forward with an ambitious timeframe". The next step will be a meeting of senior negotiators in Geneva on Oct 15-16, she said.
British Foreign Secretary William Hague said there had been a "big improvement in the tone and spirit" from Iran compared with representatives of the previous Iranian government.
German Foreign Minister Guido Westerwelle said the meeting had taken place in a "completely different tone, atmosphere and spirit" that what the group was used to and that a "window of opportunity has opened" for a peaceful resolution of the situation. He warned, though, that Iran's words would have to be matched by actions.
"Words are not enough," he said. "Actions and tangible results are what counts. The devil is in the detail, so it is now important that we have substantial and serious negotiations very soon."
Iranian President Hassan Rouhani and Zarif, both in New York this week to attend the UN General Assembly, have said they are anxious to clinch an agreement quickly that could bring their country relief from punishing international sanctions.
But the US insists Rouhani must back up his calls for moderation with actions that verify Iran is not seeking to develop a nuclear weapon.
Encouraged by signs that Rouhani will adopt a more moderate stance than his predecessor Mahmoud Ahmadinejad, but sceptical that the country's all-powerful supreme leader will allow a change in course, President Barack Obama has directed Kerry to lead a new outreach and explore possibilities for resolving the long-standing dispute.
Rouhani's pronouncements at the UN have raised guarded hopes that progress might be possible. But they have also served as a reminder that the path to that progress will not be quick or easy.
In his speech to world leaders at the UN on Tuesday, he repeated Iran's long-standing demand that any nuclear agreement must recognise the country's right under international treaties to continue enriching uranium.
The US and its allies have long demanded a halt to enrichment, fearing Tehran could secretly build nuclear warheads. They have imposed sanctions over Iran's refusal to halt enrichment. Uranium enriched to low levels can be used as fuel for nuclear energy but at higher levels, it can be used to make a nuclear weapon.
Rouhani also insisted that any deal be contingent on all other nations declaring their nuclear programs, too, are solely for peaceful purposes - alluding to the US and Israel.
Those conditions underscored that there is still a large chasm to be bridged in negotiations.
Rouhani has made a series of appearances and speeches since arriving in New York and has held bilateral negotiations with France, Turkey and Japan among others.
He has come across as a more moderate face of the hard-line clerical regime in Tehran and appears to be trying to tone down Ahmadinejad's caustic rhetoric against Israel - a point of friction in relations with the US.
On Thursday, he called for worldwide disarmament of nuclear weapons as "our highest priority."
"No nation should possess nuclear weapons, since there are no right hands for these wrong weapons," he told the first-ever meeting of a UN forum on nuclear disarmament. He was speaking on behalf of the Non-Aligned Movement, an organisation of mostly developing countries.
He repeated the organisation's long-standing demand that Israel join the international treaty banning the spread of nuclear weapons.
Israel, which has repeatedly accused Iran of aspiring to build a nuclear bomb is the only Mideast state that has not signed the landmark 1979 Non-Proliferation Treaty.
Israel reacted angrily to the remarks.
"Iran's new president is playing an old and familiar game by trying to deflect attention from Iran's nuclear weapons program," said Intelligence and International Affairs Minister Yuval Steinitz. "The problem of the NPT in the Middle East is not with those countries which have not signed the NPT, but countries like Iran, Iraq, Libya and Syria which have signed the treaty and brazenly violated it," he added.
"Unlike Iran, Israel has never threatened the destruction of another country," he said.
Rouhani also met Japanese Prime Minister Shinzo Abe, who urged "concrete progress" on settling the nuclear issue. Abe also said the window of opportunity "will not be open forever," according to Japanese Foreign Ministry assistant press secretary Masaru Sato.
Rouhani responded that he aimed to settle the nuclear issue at "an early juncture," Sato said. It was the first meeting between leaders of Japan and Iran since 2008.
Rouhani was elected after a campaign that pledged to seek relief from the international sanctions. He has welcomed a new start in nuclear negotiations in hopes they could ease the economic pressure.
He has said he has the full support of Iranian Supreme Leader Ayatollah Ali Khamenei, who has the final word on all important matters of state, including the nuclear question.
Zarif, who Rouhani has designated his lead nuclear negotiator, has urged step-by-step compromises to advance the negotiations.
Iran watchers say Rouhani may have limited time to reach a settlement - possibly a year or less - before Khamenei decides negotiations are fruitless. That may explain why Zarif has call to reach a deal in shortest timeframe possible.
Already, Iran's powerful Revolutionary Guard force has grown increasingly uneasy over Rouhani's outreach to the West as well as his apparent backing from Khamenei, who has told the Guard to steer clear of politics.
The Guard has warned Rouhani about moving too fast on his overtures.
- AP
