"In a single move, the United States is dealing a critical blow to Iran's entrenchment in Syria, Lebanon, Gaza, Iraq and Yemen".
On Sunday, Mr. Pompeo affirmed the worldwide community's backing of Mr. Trump's efforts against Iran.
After the US' withdrawal from the deal, Trump signed fresh sanctions against Iran and warned countries against any cooperation with Tehran over its controversial nuclear weapons programme.
Mr Rouhani was defiant as the sanctions kicked in, vowing Iran would continue to sell its oil.
Over the weekend, Prime Minister Benjamin Netanyahu thanked Trump for taking "historic action" against the "murderous terror regime that endangers the entire world".
The countries will deposit Iran's revenue in an escrow account, USA officials have said.
Iran is already in the grip of an economic crisis. Most Europeans, as well as Japan and South Korea, have stopped.
The US stance has already inflicted serious pain on Iranians, with the country's currency, the rial, losing more than two thirds of its value since May.
"We will proudly break the sanctions", he said at a meeting of economic officials.
Iranian state television aired footage of air defence systems and anti-aircraft batteries in two-day military manoeuvres which got under way across a vast stretch of the country's north. The waiver would be granted for six months period and then would be re-evaluated by the United States, Special US Representative on Iran, Brian Hook told reporters.
Later, a senior State Department official told reporters that that the USA was ready for talks with Iran.
The other parties to the nuclear deal see it as an important bulwark against the risk of wider war in the Middle East and have reaffirmed their commitment to it. Iran has said it could leave the pact if the European Union cannot protect its economic benefits. Selling notably accelerated on Thursday after US West Texas Intermediate crude futures fell below US$65 a barrel, a level that had stood firm in previous sell-offs during the summer and autumn.
U.S. Secretary of State Mike Pompeo, who announced the decision, did not name the eight, which he referred to as "jurisdictions", a term that might include importers such as Taiwan which the United States does not regard as a country.
The Iran nuclear deal was drawn up by the Obama administration and was agreed to by the UK, Russia, France, China, Germany and the EU.
Brian Hook, the special adviser to the secretary of state and special representative for Iran, told PBS News that the USA is not looking to give any exceptions or waivers to its sanctions regime.
Turkey said on Saturday that Ankara had received initial indications from Washington that it would be granted a waiver, but is awaiting clarification on Monday.