Iran and America exchanged rare words of conciliation on Wednesday as the Revolutionary Guard Corps released 10 US sailors who had entered sensitive waters in the Gulf.

A new crisis had seemed possible on Tuesday when the crews of two US Navy patrol boats fell into Iranian hands. Instead, Iran and America moved quickly to defuse the situation – although Tehran could not resist humiliating its captives by broadcasting pictures of the US personnel kneeling and with their hands in their heads.

One US sailor was shown apparently voicing words of contrition. “It was a mistake that was our fault and we apologise for our mistake,” he said.

But the nine men and one woman were released after one night in captivity on Farsi, an Iranian island in the Gulf. John Kerry, the US secretary of state, responded by expressing his “gratitude”, although the State Department adamantly denied Iran’s claim that he had also apologised.

Iran is scaling back its nuclear programme in accordance with the agreement signed in Vienna last July. In the coming days, the International Atomic Energy Agency (IAEA) is expected to confirm that Iran has kept its main commitments, whereupon America and its allies will lift their nuclear-related sanctions.

At least $55 billion (£36 billion) of Iranian assets will be unfrozen. As this moment approaches, neither side has an interest in derailing the process.

The US patrol boats were travelling from Kuwait to Bahrain on Tuesday when they strayed into Iranian territorial waters around Farsi Island, apparently because of mechanical and navigational failure.

Measuring less than three square miles, Farsi serves as a base for the naval wing of Iran’s Revolutionary Guard Corps in the centre of the Gulf. The US boats were intercepted at sea and the nine men and one woman on board taken into custody on the island.

The Pentagon said the US team had entered Iranian waters by mistake. In the past, Iran would have treated this with grave suspicion.

But Rear Admiral Ali Fadavi, the commander of the naval branch of the Revolutionary Guard, publicly accepted America’s explanation. “We have concluded that passage of Americans in our territorial waters was not a hostile passage or for espionage or similar acts,” he told state television. “They were positioned in that area due to the failure of their navigation systems and they were not aware of being close to Farsi Island.”

General Ali Fadavi, the Navy chief of Iran’s powerful Revolutionary GuardGeneral Ali Fadavi, the Navy chief of Iran’s powerful Revolutionary Guard Photo: APTN

The sailors were captured hours before President Barack Obama gave his final state of the union address. He did not change his text to refer to the situation, something that infuriated Republicans.

“President Obama completely omitted this latest example of Iran’s provocative behaviour so as not to interfere with his delusional talking points about his dangerous nuclear deal,” said Senator John McCain.

Other images released in the Iranian media purported to show the American personnel being held together in a room furnished with carpets and cushions. The Pentagon confirmed that all were freed unharmed at 8.43am GMT on Wednesday.

Pictures posted on Tasmin News which reportedly show the US Navy personnel detained by Iran’s Revolutionary GuardsPictures posted on Tasmin News which reportedly show the US Navy personnel detained by Iran?s Revolutionary Guards

The handling of the US sailors contrasted sharply with treatment meted out to 15 British naval ratings and marines who were captured by Iran in the Gulf in 2007: