The US Treasury said the sanctions would freeze any of Zaria's assets in the United States or controlled by US entities, as well as squeeze his ability to function as a globe-trotting diplomat.
"They are afraid of our foreign minister's interviews," Rouhani said in a televised speech, referring to a recent round of interviews Zarif gave to foreign media in New York.
"It is completely clear that the foundations of the White House have been shaken by the words and logic of an informed, devoted and diplomatic individual.
"They are doing childish things now. Maybe there's no better way to describe (the sanctions) but childish," Rouhani said on a visit to the northwestern city of Tabriz.
"Our enemies are so helpless that they have lost the ability to act and think wisely."
The designation of Zarif under the same sanctions earlier applied to Iran's supreme leader Ayatollah Ali Khamenei is the latest in a string of moves by the United States against the Islamic republic.
"Zarif implements the reckless agenda of Iran's supreme leader, and is the regime's primary spokesperson around the world," Treasury Secretary Steven Mnuchin said in a statement.
"The United States is sending a clear message to the Iranian regime that its recent behaviour is completely unacceptable."
The arch-foes have been locked in a battle of nerves since President Donald Trump withdrew the US from a landmark 2015 deal aimed at limiting Iran's nuclear programme and began reimposing sanctions.
The situation has worsened since the Trump administration stepped up a campaign of "maximum pressure" against Iran this year, with drones downed and tankers mysteriously attacked in Gulf waters.
- 'Iran's primary spokesperson' -
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But in a mixed message to Tehran, Washington on Wednesday extended waivers for three Iranian civil nuclear projects, to avoid upsetting the other signatories to the 2015 Iran nuclear agreement -- China, Russia, Germany, France and Britain.
"This is a short 90-day extension," said White House national security adviser John Bolton, a champion of the hawkish policy towards Iran.
"We are watching those nuclear activities very, closely, they remain under daily scrutiny," he told Fox Business.
Zarif has been at the heart of complex talks with foreign capitals over Iran's nuclear power industry, which Tehran says is peaceful, but Washington and regional allies including Israel insist is cover for a secret weapons programme.
But a senior Trump administration official said Zarif's diplomatic image -- bolstered by his fluent English, self-effacing humour and background as a US-educated academic -- was false.
"The key issue is that he has had this veneer... of being the sincere and reasonable interlocutor for the regime. Our point today is that he is no such thing," the official said on condition of anonymity.
"Today President Trump decided enough was enough," the official said, accusing Zarif of functioning as "propaganda minister, not foreign minister".
Zarif shot back, tweeting that the United States was trying to silence Iran on the international stage.
"The US' reason for designating me is that I am Iran's 'primary spokesperson around the world' Is the truth really that painful?" he wrote.
"We know that calling for dialog & peace is an existential threat to #B_Team," he added, referring to Bolton, Israeli Prime Minister Benjamin Netanyahu and others pushing a hard line on Iran.
Officials in Iran rallied around the foreign minister on Thursday.
"America is not only afraid of Iran's missiles but also of Iran talking," Abbas Ali Kadkhodayi, spokesman for Iran's law-vetting Guardian Council, said on Twitter.
"Sanctioning @JZarif means America's claim of freedom of speech is false. Sanctioning Zarif means the fall of the Statue of Liberty."