It is the first time the US has labelled another nation’s military in this way.
President Hassan Rouhani said the US was the “leader of world terrorism”.
Washington-Tehran tensions have risen since Mr Trump withdrew the US from the international Iran nuclear pact.
Iran’s government took retaliatory action against the US move by naming US Central Command (Centcom) as a terrorist organisation and the US government as a sponsor of terror.
Centcom is the Pentagon wing that oversees Washington’s security interests across the central area of the world map, most notably Afghanistan, Iraq, Iran, Pakistan and Syria.
What did Iran’s leaders say?
President Rouhani condemned the US move in a speech broadcast live on state television.
“Who are you to label revolutionary institutions as terrorists?” he asked. “You are the leader of world terrorism.”
“This mistake will unite Iranians and the Guards will grow more popular in Iran and in the region.
“America has used terrorists as a tool in the region while the Guards have fought against them from Iraq to Syria.”
Supreme Leader Ayatollah Ali Khamenei said the IRGC had confronted enemies at home and abroad, state TV reported.
They were “in the frontline of confronting enemies of our revolution and had always defended the country… America has failed to block our advancements”.
Why has the US done this?
Labelling the Guards as a terrorist organisation will allow the US to impose further sanctions – particularly affecting the business sector, given the IRGC’s involvement in Iran’s economy.
A number of IRGC and affiliated entities have already been targeted by US sanctions for alleged proliferation activities, support for terrorism and human rights abuses.
Mr Trump’s statement on Monday said: “This unprecedented step, led by the Department of State, recognises the reality that Iran is not only a state sponsor of terrorism, but that the IRGC actively participates in, finances, and promotes terrorism as a tool of statecraft.”
The president added that the move was meant to “significantly expand the scope and scale” of pressure on Iran.
“If you are doing business with the IRGC, you will be bankrolling terrorism,” Mr Trump said.
The measure will take effect early next week, according to the state department.
US Secretary of State Mike Pompeo and National Security Adviser John Bolton, both Iran hawks, championed the decision.
BBC defence correspondent Jonathan Marcus says that few Western commentators would disagree that the IRGC is responsible for all sorts of disruptive activities in the region and beyond.
But he adds that many – including it seems some officials in the state department and the Pentagon – fear that this step could simply backfire as it could encourage the IRGC or its proxies to take action against US personnel or other targets in places where they might be vulnerable, for example in Iraq.
What is the IRGC?
Iran’s most elite military unit, the IRGC was set up shortly after the 1979 Iranian revolution to defend the country’s Islamic system, and to provide a counterweight to the regular armed forces.
It has since become a major military, political and economic force in Iran, with close ties to the Supreme Leader, Ayatollah Ali Khamenei, and many other senior figures hailing from its ranks.
The IRGC is estimated to have more than 150,000 active personnel, boasts its own ground forces, navy and air force, and oversees Iran’s strategic weapons, including its ballistic missiles.
The IRGC exerts influence elsewhere in the Middle East by providing money, weapons, technology, training and advice to allied governments and armed groups through its shadowy overseas operations arm, the Quds (Jerusalem) Force.