Mohammed bin Salman said failure to act could embolden Iran and lead to war, which he said would ruin the global economy, following an attack on oil facilities which he blames on Tehran.
Iran said the prince’s remarks would “bring [the Saudis] nothing but shame”.
The prince also said he accepted some responsibility for journalist Jamal Khashoggi’s killing.
But, speaking to CBS News, he denied personally ordering it.
The prince, who is considered the de facto ruler of Saudi Arabia, is suspected of personally targeting Mr Khashoggi, a Saudi journalist critical of the government in Riyadh.
Mr Khashoggi was killed in Saudi Arabia’s consulate in Turkey on 2 October 2018.
In an interview with CBS’ 60 Minutes news programme on Sunday, he said: “I take full responsibility as a leader in Saudi Arabia, especially since it [the killing] was committed by individuals working for the Saudi government.”
However, he denied ordering the killing of Mr Khashoggi directly, or having knowledge of it at the time.
Saudi authorities have since blamed a “rogue” operation for his murder and put 11 men on trial.
The prince also appeared to offer talks for a political solution to the civil war in Yemen, where government forces backed by a Saudi-led coalition are fighting Houthi rebels supported by Iran.
Iran is Saudi Arabia’s regional rival and an opponent of the US, which pulled out of a treaty aimed at limiting Tehran’s nuclear programme after Mr Trump took power.
US-Iran tensions have risen markedly this year, with the US blaming Iran on attacks on six oil tankers in the Gulf between May and July. Tehran rejects the accusations.