Meanwhile, EU states that backed the nuclear deal have said they will protect EU companies doing “legitimate” business with Iran.
President Trump withdrew from the agreement in May, describing it as “defective at its core” because it had not stopped Iran developing a ballistic missile program and intervening in neighboring countries.
The president tweeted after the announcement: “Sanctions are coming,” referencing the TV series Game of Thrones and its motto “Winter is coming”.
The US has been gradually re-imposing sanctions, but analysts say this move is the most important because it targets the core sectors of Iran’s economy.
The agreement saw Iran limit its controversial nuclear activities in exchange for sanctions relief.
The US sanctions will cover shipping, shipbuilding, finance and energy.
More than 700 individuals, entities, vessels and aircraft will be put on the sanctions list, including major banks, oil exporters and shipping companies.
Treasury Secretary Steven Mnuchin also said that the Brussels-based Swift network for making international payments was expected to cut off links with targeted Iranian institutions.
Being disconnected from Swift would almost completely isolate Iran from the international financial system.
They are the second lot of sanctions re-imposed by President Trump since May.
Secretary of State Mike Pompeo set out 12 demands that Iran must meet if sanctions are to be lifted – including ending support for militants and completely ballistic missile development.
Mike Pompeo did not name the eight countries that been granted waivers to continue importing Iranian oil.
The secretary said the eight had “demonstrated significant reductions in their crude oil and co-operation on many other fronts”. Two would eventually stop imports and the other six greatly reduce them, he added.
US allies such as Italy, India, Japan and South Korea are among the eight, the Associated Press reports. Turkey also obtained a waiver, Reuters says.
All have been have been top importers of Iranian oil.
Iran’s foreign ministry spokesman Bahram Qasemi told state TV that Iran had “the knowledge and the capability to manage the country’s economic affairs”.
Fresh sanctions have been imposed on Iranian companies and individuals by the US over a recent ballistic missile test.
The new sanctions prevent 11 entities and individuals linked to the missile program from using the US banking system.
The move came after international nuclear sanctions on Iran were lifted as part of a deal hailed by President Barack Obama on January 17 as “smart”.
Four American-Iranians were also freed in a prisoner swap as part of the deal.
Among them was Washington Post reporter Jason Rezaian – whom President Barack Obama described as “courageous”. A fifth American was freed separately.
Jason Rezaian and two of the others freed flew to a US base in Germany via Geneva for medical evaluation.
Another, Nosratollah Khosravi-Roodsari, did not fly out with the others, US officials said. A fifth man, Matthew Trevithick, was freed in a separate process.
Meanwhile the US said it had offered clemency to seven Iranians being held in the US for sanctions violations.
Negotiations in December over the prisoner exchange delayed the US Treasury’s imposition of the latest sanctions.
They were only announced once the plane containing the former prisoners had left Iran, reports said.
They were triggered by Iran conducting a precision-guided ballistic missile test capable of delivering a nuclear warhead last October, violating a UN ban.
“Iran’s ballistic missile program poses a significant threat to regional and global security, and it will continue to be subject to international sanctions,” said Adam J. Szubin, US acting under-secretary for terrorism and financial intelligence.
Moments later, President Barack Obama hailed the nuclear deal, which is being implemented following verification by the International Atomic Energy Agency (IAEA) that Iran had restricted its sensitive nuclear activities.
“This is a good day because once again we’re seeing what’s possible with international diplomacy,” he said.
“For decades our differences meant our governments almost never spoke – ultimately, that did not advance America’s interests,” Barack Obama said.
The deal meant “Iran will not get its hands on a nuclear bomb”, he said.
Barack Obama said differences with Iran remained, and the US would “remain steadfast in opposing Iran’s destabilizing behavior elsewhere” – such as its missile tests.
The president defended a separate settlement at an international legal tribunal which will see the US repay Iran $400 million in funds frozen since 1981 plus a further $1.3 billion in interest – saying there was no point “dragging this out”.
Earlier, Iranian President Hassan Rouhani said the nuclear deal opened a “new chapter” in the country’s relations with the world.
Iran nuclear deal has been welcomed by many governments, the UN and EU – but disparaged by some US Republicans and Israel, which says it allows Iran to continue to “spread terror”.
The economic sanctions on Iran have been lifted after the country’s compliance with obligations under its nuclear agreement with world powers was certified.
The EU foreign policy chief, Federica Mogherini, announced the lifting of sanctions in accordance with the deal “as Iran has fulfilled its commitment”.
International nuclear watchdog IAEA said its inspectors had verified that Iran had taken the required steps.
A deal between Iran and world powers was agreed in July 2015.
The IAEA’s Director General Yukiya Amano said it was “an important day for the international community”.
Lifting the international sanctions on Iran will unfreeze billions of dollars of assets and allow Iranian oil to be sold internationally.
As part of the deal, Iran had to drastically reduce its number of centrifuges and dismantle a heavy-water reactor near the town of Arak, both of which could be used in creating nuclear weapons.
Iran has always maintained its program is peaceful, but opponents of the deal – such as some US Republicans – say it does not do enough to ensure the country cannot develop a nuclear bomb.
Secretary of State John Kerry has ordered that US nuclear-related economic sanctions against Iran be lifted.
Iran’s President Hassan Rouhani congratulated the nation following the IAEA announcement. The deal “had “borne fruit,” Hassan Rouhani tweeted, adding that he took “a bow before a patient nation like ours”.
On the same day it emerged that Iran had released Washington Post reporter Jason Rezaian and three other Iranian-American prisoners in an apparent prisoner swap with the US.
Jason Rezaian, 39, was jailed on charges, including espionage, in November 2015.
America said it was offering clemency to seven Iranians being held in the US for sanctions violation.
Boeing has received permission from the US Treasury to export certain spare commercial parts to Iran, a company spokesman says.
Boeing has had no public dealings with Tehran since 1979.
In a statement, the plane maker said the license had been granted for the safety of flight.
Iran Air is still flying passenger planes bought before the 1979 hostage crisis
The step is being seen as part of a temporary agreement to ease sanctions on Tehran that US Secretary of State John Kerry reached with Iran last year.
Under the deal brokered in November, Iran agreed to curtail its nuclear activities for six months in exchange for sanctions relief from nations including Britain, China and the US.
General Electric said late on Friday it had received US permission to overhaul 18 engines sold to Iran in the late 1970s. That work would be carried out at GE facilities or at German firm MTU Aero Engines, it said.
Iran Air is still flying passenger planes bought before the 1979 hostage crisis, during which 52 Americans were held hostage in Tehran for 444 days.
Iran has reportedly argued that sanctions imposed after the hostage ordeal have prevented Tehran from upgrading its plane fleet and reduced the safety of its aircraft.
There have been more than 200 accidents involving Iranian planes in the past 25 years, leading to more than 2,000 deaths, reports say.
Boeing has said the license covers only components required to ensure ongoing safe flight operations of planes it sold before Iran’s revolution in 1979.
No discussions are to be allowed over the sale of new aircraft when and if sanctions are completely lifted, correspondents say. If a permanent deal is agreed, it is thought likely that Iran would require the purchase of hundreds of new aircraft.