In a live televised address, Rouhani said "God has accepted the nation's prayers" with the deal, which could open up grounds for greater international cooperation.
It would lift what he called "inhumane and tyrannical sanctions" that caused years of economic distress and a deep recession that only ended last year and high inflation which eroded people's savings and buying power.
Rouhani, elected on a pledge to solve the nuclear crisis, spoke minutes after US President Barack Obama's comments on the agreement had been broadcast live on Iranian state television.
But Obama's speech was cut off when Rouhani, who has faced hardline opposition yet staked his presidency on resolving the nuclear standoff with the West, spoke in Tehran.
"If this deal is implemented correctly... we can gradually eliminate distrust," Rouhani said, alluding to Iran's long-strained relations with leading Western states.
"This is a mutual deal, a reciprocal deal," he said, noting that "all our objectives" had been met as sanctions would be lifted and the country's civilian nuclear programme acknowledged.
- Final word -
The pact would also "take the nuclear dossier" out of the UN Security Council's remit, reversing what he called "illegal resolutions" passed by the world body against Iran.
Although Rouhani pushed hard for a nuclear deal, the final word rested with Ayatollah Ali Khamenei, Iran's supreme leader, who set tough red lines for the talks, and whom the president thanked in his speech.
Iran's parliament will review the agreement and has the task of ratifying it, but top officials have said a deal with Khamenei's endorsement will not be obstructed.
The long-sought accord came after 18 days of talks in Vienna, the culmination of 22 months of diplomacy between Iran and the P5+1 group -- Britain, China, France, Russia and the United States plus Germany.
In a message posted on Twitter before the deal was formally unveiled in Vienna, Rouhani said "new horizons" could open now that "this unnecessary crisis" has been resolved.
There can now be "a focus on shared challenges", he added, alluding to Sunni Muslim extremists of the Islamic State group, who from their base in Iraq and Syria are launching attacks on both Shiite and Western targets worldwide.
Rouhani's tweet came shortly after Foreign Minister Mohammad Javad Zarif announced at talks in Vienna: "We are starting a new chapter of hope."
Since his election in 2013, Rouhani had sought to end the 13-year international standoff over Iran's controversial nuclear programme -- and with it the sanctions that have paralysed the economy.
- Strict limits -
In office, Rouhani has cut inflation from 42 percent to 15 percent and achieved growth of 3 percent last year.
The deal puts strict limits on Tehran's nuclear activities for at least a decade and calls for stringent UN oversight. World powers say this will make any dash to an atomic bomb virtually impossible.
Iran has always denied Western suspicions that it has been trying to acquire the know-how to make nuclear weapons, and Rouhani reiterated that stance Tuesday.
"Iran will never seek a nuclear weapon, with or without the implementation" of the Vienna deal, he said, adding that to do so would contradict a fatwa from Khamenei.
Wider political reaction was muted in Tehran.
However, Hossein Marashi, a former MP and member of the country's reformist movement, praised Rouhani and Khamenei's leadership on the nuclear issue.
"Experience has shown that whenever the leadership focuses on an issue we've managed to resolve it," Marashi said.
"We managed to sit at the negotiating table with rival powers who had long enmity against Iran. If such a focus happens in domestic politics, many of our problems will be resolved."
But Raja, a hardline news website, railed against celebrating the deal, describing it as "not a victory but a definite loss" as the accord had "shut down a substantial part of Iran's nuclear technology".
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