The historic 15-minute call between the US president and his Iranian counterpart on Friday marked the first contact between leaders of the two nations in over three decades -- since the 1979 Islamic revolution.
It heralded a "new tone" in relations, said Shashank Joshi, research fellow at the Royal United Services Institute think tank in London.
Stateside, the optimistic New York Times wrote: "It’s hard not to be swept up in the euphoria, especially when an adversary begins to seem not only reasonable but personable."
But now begins the hard task for Obama and Rouhani of persuading hostile diplomats and advisers that normalising relations is desirable.
Tehran's nuclear programme and the opposition of Israel, the US ally and Iran foe, are likely to make full rapprochement difficult.
"Hardliners on both sides are the big elephant in the room," said Arshin Adib-Moghaddam, chair of the Centre for Iranian Studies at London's School of Oriental and African Studies.
"Rouhani has to tread a fine line," he warned. "The issue of relations with the United States is emotive, charged with historical memories that are personal and immediate. There will be situations when he has to backtrack because of domestic constraints."
Obama and Rouhani risk the wrath of their respective entourages and allies as they move to restore diplomatic relations which broke down after the hostage-taking at the US embassy following the 1979 revolution.
"For Obama this will open him up to criticism from Israel and hardliners in Washington. He is taking a brave step in doing this," Joshi said.
"Rouhani is also taking a brave step as we've seen from the demonstrations upon his return. He will have angered many, many hardliners," he added.
Signup to our newsletter and follow us on Facebook and Twitter!
As Rouhani arrived back in Tehran from the UN General Assembly in New York, a protester threw a shoe at his motorcade while others in a crowd of several dozen chanted "Death of America".
But another larger group of supporters turned out to thank Iran's new president -- a moderate who replaced the firebrand Mahmoud Ahmadinejad in June -- for taking the important step. The Iranian press has also lauded the renewed contact.
Keen to assuage fears that rapprochement will see Tehran soften its stance on its nuclear ambitions, Rouhani said he had stood firm on Iran's "rights" and "objectives" and defended the Iranian position in New York in line with guidance from supreme leader Ayatollah Ali Khamenei.
"The supreme leader has given him the political authority to pursue this dialogue to at least some degree," Joshi said.
The West wants major concessions from Iran including the suspension of all enrichment of uranium beyond the level required to fuel nuclear power plants, and the closure of Iran's underground enrichment facility near the central city of Qom.
"The evidence suggests Rouhani is empowered to act on the nuclear dispute," Joshi said.
"It doesn't suggest that Rouhani has been given the authority to negotiate a full-scale rapprochement, on issues like Syria, Lebanon, the Persian Gulf, on Iran's alleged meddling in Yemen. The issues that divide these countries go well beyond the nuclear dispute."
In Israel, a displeased Benjamin Netanyahu accused Rouhani of "playing for time" to plough on with Iran's nuclear programme, ahead of what could now be a tense visit by the prime minister to the White House on Monday.
French Foreign Minister Laurent Fabius praised "a very, very clear change of attitude" from the Iranians while cautioning that the world must wait "to see how things change in reality".
"The Iranian press is already talking about the creation of an Iranian-American chamber of commerce," Thierry Coville from the Iris research institute told AFP.
But Iran's relationship with Europe is also key, he said. "Europe also has to know how to renew ties with Iran and defend its interests."