The first of two rounds of US sanctions kicked in at 12:01 am (0431 GMT), targeting Iran's access to US banknotes and key industries, including cars and carpets, AFP reported.
Trump's contempt for the nuclear deal dates back to his time as presidential candidate and on May 8, he made good on a pledge to pull America out of the international agreement.
The unilateral withdrawal came despite other parties to the agreement -- Britain, China, France, Germany, Russia and the EU -- pleading with Trump not to abandon the pact, highlights the US leader's go-it-alone style and his distaste for multilateral agreements.
In an executive order Monday, Trump claimed the sanctions seek to pile financial pressure on Tehran to force a "comprehensive and lasting solution" to Iranian threats, including its development of missiles and regional activities.
The European Union's diplomatic chief Federica Mogherini said the bloc, as well as Britain, France and Germany, deeply regretted Washington's move.
"We are determined to protect European economic operators engaged in legitimate business with Iran," she said in a statement.
Many large European firms are leaving Iran for fear of US penalties, and Trump warned of "severe consequences" against firms and individuals that continued to do business with Iran.
Trump said he was open to reaching a more comprehensive deal with Iran.
But Iranian President Hassan Rouhani was unimpressed by the offer.
"If you're an enemy and you stab the other person with a knife, and then you say you want negotiations, then the first thing you have to do is remove the knife," the Iranian leader said in an interview on state television.
"They want to launch psychological warfare against the Iranian nation," Rouhani said. "Negotiations with sanctions doesn't make sense."
John Glaser, director of foreign policy studies at the Cato Institute, noted that the US sees the sanctions "as a tool to pressure Iran to come back to the negotiating table to rehash the nuclear deal on terms more to Trump's liking. That is not going to happen."
The second phase of US sanctions, which takes effect November 5 and will block Iran's oil sales, is due to cause more damage, though several countries including China, India and Turkey have indicated they are not willing to entirely cut their Iranian energy purchases.
Iran's Foreign Minister Mohammad Javad Zarif told reporters the global reaction to Trump's move showed that the US was diplomatically "isolated," but acknowledged the sanctions "may cause some disruption."