The two vessels were struck by explosions in the early daylight hours Thursday after leaving the Strait of Hormuz and traveling around 25 nautical miles off Iran's southern coast headed toward Asia.
The 23 crew members on Front Altair were rescued by a nearby vessel then transferred to an Iranian navy boat and disembarked at a local Iranian port.
Taiwan's state oil refiner CPC said the Front Altair, owned by Norway's Frontline, was "suspected of being hit by a torpedo" around 04:00 GMT, as it carried 75,000 tonnes of the petrochemical feedstock naphtha to Taiwan.
According to a US official, initial evidence suggested the attack against the Kokuka Courageous was conducted by Iran with a mine similar to what was used against oil tankers off the UAE last month.
Another tanker, the Norwegian-owned 'Front Altair, ' also suffered some damage while passing through the Gulf of Oman at around the same time.
The attacks, which came a month after a similar incident targeting four tankers off the coast of the United Arab Emirates, "demonstrate the clear threat that Iran poses to worldwide peace and security", Mr Cohen said.
Washington has also accused Tehran of being behind a similar attack on May 12 on four tankers in the same area, which is a vital shipping route through which much of the world's oil passes.
Iran has rejected the United States accusation against it as "baseless".
The kingdom, the world's top oil exporter, ratcheted up tensions with Iran after the attacks off Fujairah, which were followed by a drone strike on a key Saudi oil pipeline claimed by Yemen's Iran-aligned Huthi rebels.More news: Oil Rises As Russia Looks to Support Continued Production Cuts
More news: Ford recalls 1.3 million vehicles for suspension, transmission woes
More news: Russian police drop charges against investigative reporter Golunov after outcry
About an hour after the Kokuka Courageous' distress call, a U.S. aircraft observed an IRGC Hendijan-class patrol boat and multiple IRGC fast attack craft and fast inshore attack craft near the Altair, Capt Urban said.
Khamenei told Abe on Thursday that it was pointless to reply to a message he had brought to Tehran from Trump. Iran called on "the global community to live up to its responsibilities in preventing the reckless and risky policies and practices of the USA and its regional allies in heightening the tensions in the region".
Tensions between the USA and Iran have been rising since President Donald Trump pulled out of Barack Obama's nuclear deal past year, which traded sanctions relief for certain curbs on the regime's nuclear weapons program. He described the ships as being hit in a "reported attack", without elaborating.
Tensions have escalated in the Mideast as Iran appears poised to break the 2015 nuclear deal with world powers, an accord that President Donald Trump repudiated a year ago.
Already, Iran says it quadrupled its production of low-enriched uranium. Japan's Trade Ministry said the two vessels had "Japan-related cargo", without elaborating. "Some experts point out that the conflict might be triggered accidentally", he said.
The attacks came as Japanese Prime Minister Shinzo Abe was in Iran seeking to defuse tensions between Washington and the Islamic republic.
"We have no doubt about your good will and seriousness, but ..." But Ayatollah Khamenei dismissed the notion of talks saying in a series of tweets "I do not consider Trump as a person worth exchanging any message with and I have no answer for him, nor will I respond to him in the future".
It is unknown who would benefit from targeting the ships, but the timing of the incidents is especially sensitive.
Iranian search and rescue teams picked up 44 sailors from the two damaged tankers and took them to the Iranian port of Jask, Iran's IRNA reported.
Associated Press writers Aya Batrawy in Dubai, David Rising in Berlin, Mari Yamaguchi in Tokyo, Lolita C. Baldor in Washington and Amir Vahdat in Tehran, Iran, contributed.