President Barack Obama faced a bruising battle to sell the deal in Washington as Congressional leaders queued up to denounce it.
In return for curbs on its nuclear programme for at least 10 years, Iran will be freed from Western and UN sanctions that have crippled its economy.
Foreign Minister Mohammad Javad Zarif, who led Iran's negotiating team in the final 18 days of talks that culminated in yesterday's deal, said on his return home that common ground had been found with the six powers led by the US.
"It will happen in around four months from now," he said of the deal's formal implementation.
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Zarif's comments came after a night of celebrations in Tehran, where his own name was chanted in the streets by joyous Iranians.
Many festooned their cars with balloons and danced in the street to celebrate the prospect of an end to long years of economic hardship caused by Western sanctions.
"Maybe the economy is going to change, especially for the young people. I was thinking about leaving, but now I will stay to see what happens," said Giti, 42, a computer programmer.
Financial daily Donyaye Eqtesad said Iran had "entered the post-sanctions age."
"Iran Siege Broken," headlined the moderate daily Ghanoon.
In Washington, however, thedeal was intensely scrutinised.
The speaker of the Republican-led US House of Representatives, John Boehner, said it was "likely to fuel a nuclear arms race around the world".
But Zarif hit back at the deal's biggest critic, Israeli Prime Minister Benjamin Netanyahu, head of the region's sole if undeclared nuclear state, who branded the agreement a "historic mistake".
"Netanyahu kicked up a fuss, as he is upset that Iran managed to get sanctions lifted and prevent a manufactured crisis," Zarif said.
Iran has always denied seeking an atomic bomb, a stance President Hassan Rouhani reiterated after yesterday's deal.