He made the comments in an interview with CBS on Sunday, during his two-and-a-half-week tour of the US.
A spokesman from Iran's foreign ministry, Bahram Qassemi, dismissed Bin Salman's comments as "big lies" on Tuesday, and described Saudi Arabia's role in creating and funding terrorist organizations across the globe as "undeniable."
"Bin Salman in his visit to the US is heinously trying to pave the way for his coronation by any means, but it seems that pouring the hundreds of billions of dollars of the Saudi people into the arms industry of the US has not served the purpose, and he is now trying to purge the proved records of current rulers of Saudi Arabia on supporting terrorism and extremism," he said.
Spokesman Qassemi explained that some members of al-Qaeda fled to Iran during the US-led invasion of Afghanistan, but insisted that they were dealt with accordingly by Iranian authorities and law enforcement agencies.
"They were arrested and extradited to their respective governments, according to their documents and nationalities, including a number of [Osama] Bin Laden's family members, that because of their Saudi nationalities, Riyadh was informed. Following the coordination, Bin Laden's daughter was extradited to Saudi embassy in Tehran."
In the same interview with CBS on Sunday, the Saudi Crown Prince also said that "Iran is not a rival to Saudi Arabia," and went on to boast about Saudi Arabia's economic superiority over Iran.
"Its (Iran's) army is not among the top five armies in the Muslim world. The Saudi economy is larger than the Iranian economy. Iran is far from being equal to Saudi Arabia," the prince said.
Saudi Arabia and Iran are backing opposing sides in a number of regional conflicts — including the ongoing war in Syria — and relations look set to worsen, especially with Mike Pompeo, a harsh critic of the Iran nuclear deal, now US secretary of state.