In a tweet written on the anniversary of the upheaval that was also sent out in Farsi, Trump said: "40 years of corruption. 40 years of repression. 40 years of terror. The regime in Iran has produced only #40YearsofFailure.
"The long-suffering Iranian people deserve a much brighter future," he added.
Earlier, Trump's chief foreign policy advisor John Bolton issued a similar statement, tweeting that "it's been 40 yrs of failure. Now it's up to the Iranian regime to change its behaviour, & ultimately up to the Iranian people to determine the direction of their country."
Bolton said Washington would support "the will of the Iranian people, & stand behind them to ensure their voices are heard."
Bolton, a leading hawk in the Trump administration's attempt to weaken Iranian influence, was tweeting as a huge crowd in Tehran gathered to celebrate the 1979 revolution, in which Ayatollah Ruholla Khomeini ended the centuries-old rule of the royal dynasty.
The head of the Iranian regime Hassan Rouhani told Iranians they should resist a "conspiracy" involving Washington.
The two countries have not had diplomatic relations since 1980, and Trump has pulled the United States from an international agreement meant to reward Iran for giving up nuclear weapon ambitions.