As part of the new sanctions, the US Treasury Department will add more than 700 names to its list of blocked entities as of November 5, with the US working closely with countries to cut off Iranian oil exports as much as possible.
The Iranian president said: "America wanted to cut to zero Iran's oil sales. but we will continue to sell our oil.to break sanctions".
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.
All of the other signatories, China, the European Union, France, Germany, Iran, Russia and the United Kingdom, have remained in the deal despite USA action.
For example, the USA does not want to have India and South Korea move geopolitically closer to China, Russia and other competitors and away from its influence. "Essentially the sanctions that were taken off are snapping back", said Katherine Bauer, the Blumenstein-Katz Family Fellow at the Washington Institute.
"In a single move, the United States is dealing a critical blow to Iran's entrenchment in Syria, Lebanon, Gaza, Iraq and Yemen", the minister posted in a rare English-language tweet.
Asked if the USA had assurances from India and China - suspected to be among the nations getting waivers - that they would eventually give up importing oil from Iran, Pompeo wouldn't say.
"Iranian exports and production had been declining steadily in advance". But he said the eight unidentified nations "need a little bit more time to get to zero".
They have promised to support European firms that do "legitimate business" with Iran and have set up an alternative payment mechanism - or Special Purpose Vehicle (SPV) - that will help companies trade without facing United States penalties.
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'We are in the war situation, ' he said.
All of that said, as Pompeo noted on Friday, the waivers are not full waivers. However, one must keep in mind that the Iranian government has stated that it will continue to sell oil in different ways. "We think China and Iran carrying out normal cooperation under the framework of global law is lawful and reasonable, and (this right) should be respected and protected", he added.
Iran's economy has been hit hard, with the country's currency, the rial, plummeting in value, leading to some street protests in Iranian cities.
There have also been concerns about the price and supply of oil globally, meaning that without some Iranian oil the price might increase too fast - the price already having been much higher in recent months.
Pompeo and Treasury Secretary Steven Mnuchin were expected to provide more details on the sanctions at a briefing Monday.
India and South Korea do not border Iran, but they both rely heavily on its oil and they are important countries to overall USA interests.
India's top banker State Bank of India, which was handling most of the payments to Iran for oil purchases, is now complying with the USA sanctions against Tehran, its chairman said on Monday.
Iranian students stormed the embassy shortly after the 1979 Islamic Revolution, taking 52 Americans hostage for 444 days.