Yet only five days earlier, the US House of Representatives passed a sanctions bill that would further erode Iran's oil revenues.
In Iran, the Intelligence Minister prevented Rouhani from inviting to the inauguration a key supporter of his electoral campaign, the former reformist President Mohamed Khatami. Khatami is linked to the Green Movement whose leaders are being asked to "repent" for their membership of the group known as the "sedition".
Jack Straw, the former British Foreign Secretary, recommended in a BBC radio interview "confidence building" unilateral concessions to Iran by the six big powers negotiating on its nuclear programme which remains the most contentious issue between Iran and the West. The economic sanctions, targeting the banking and oil industries, were imposed over Iran's refusal to stop uranium enrichment which could be a pathway to a bomb.
Straw, who became known as "Ayatollah Jack" for his frequent visits to Tehran, dealt with Rouhani as Iran's chief nuclear negotiator in talks which culminated in a short-lived nuclear deal with the EU in 2004.
But the Israeli Prime Minister, Benjamin Netanyahu, who has warned the West against being "duped" by the new President, noted that only two days before his inauguration he had described Israel as "a wound on the body of the Islamic world". Iranian state television agencies said that Rouhani had been misquoted by a rogue news agency, reflecting the continuing factional struggle between reformists and hardliners in Tehran, but the damage was done.
"The President of Iran has been replaced, but the goal of the regime has not been replaced," said Netanyahu. "Iran's intention is to develop a nuclear capability and nuclear weapons in order to destroy the state of Israel and we are all committed to prevent this."
The West will find that Rouhani, a Western-educated career diplomat, is no pushover. But he will have limited room to manoeuvre as President. As President Khatami found to his cost, he will have to deal with the powerful judiciary and Revolutionary Guards.
Above all, on matters of national security such as Iran's nuclear programme, it is the Spiritual Leader, Ayatollah Ali Khamenei, who decides policy.
Khamenei gave Rouhani his blessing on Sunday, when he officially took over as President. But the leader's statement that Rouhani "should realise the fact that the people see Iran as an Islamic country," could also be taken as a warning.
Rouhani yesterday unveiled his Cabinet which contains reformists in key positions such as the foreign and oil ministries. Mohamed Javad Zarif, his choice as Foreign Minister, is well known in Washington as a former UN ambassador. Bijan Zanganeh, who served as oil minister under Khatami, has been asked to return to the post which he held from 1997 to 2005. Khamenei reportedly rejected Rouhani's original choices for the defence and intelligence portfolios.