Saudi Arabia has said it will carry out urgent reprisals as it accused Iran of being behind a late-night cruise missile attack by Houthi rebel fighters on a Saudi international airport that injured 26 people.

The Saudi foreign ministry said the Command of Joint Forces of the Coalition promised it “will take urgent and timely measures to deter these Iranian-backed terrorist Houthi militias”.

The attack on Abha airport was condemned across the Middle East and by the US defence department.

The Saudi-backed Yemeni government, which has been fighting a four-year civil war against the Houthi rebels, claimed the missile directed at the airport had been supplied by Iran, even claiming Iranian experts were present at the missile’s launch.

Iran strongly denies Saudi claims of aiding the Houthi movement.

The Houthi rebels insist they have a right to defend themselves from a Saudi directed blockade, and reported an initial Saudi reprisal that hit densely populated areas in the north of the country.

Diplomats will fear that the conflict in Yemen is spilling over into the dispute between Washington and Tehran, particularly if the US backs claims that Iran is directing the increasingly sophisticated Houthi attacks deep into Saudi territory.

A Houthi military spokesman promised the group would target every airport in Saudi Arabia and that the coming days would reveal “big surprises”.

No fatalities were reported in the airport attack, which hit the arrivals hall, but the number of civilians wounded was the largest in any Houthi attack inside Saudi Arabia.

The Houthis’ al-Masirah satellite news channel said the missile hit its intended target, Abha airport, near the Yemen border, disrupting flights. The rebels have also carried out drone strikes on Saudi oil installations and may have been responsible for recent attacks on oil tankers off the coast of the United Arab Emirates.

A UAE-led investigation into the shipping attacks was unable to identify the culprits, but said a state-supported actor was involved.

The precise extent to which Iran is providing military assistance to the Houthi movement is a matter of dispute, but UN reports suggest it has provided weaponry.

Iran operates through surrogates, but has looked as if it was seeking ways to reduce tensions with the US. The Japanese prime minister, Shinzo Abe, arrived in Tehran on Wednesday, carrying what Iran expects is a message on behalf of Donald Trump that sets out US conditions for direct talks.

Iran is threatening to pull out of the 2015 nuclear deal unless the US relaxes economic sanctions that are crippling the the country’s economy. Trump pulled the US out of the deal last year.

A spokesman for the Saudi-led coalition, Turki al-Maliki, was quoted by the state-run Al Ekhbariya news channel as saying three women and two children were among those hurt, and that eight people were taken to hospital while 18 sustained minor injuries.

A Houthi spokesman, Mohamed Abdel Salam, said the attack was in response to Saudi Arabia’s “continued aggression and blockade on Yemen”. Earlier in the week, he said attacks on Saudi airports were “the best way to break the blockade” of the airport in Yemen’s capital, Sana’a, which the rebels overran in late 2014. Tens of thousands of civilians have been killed in the conflict since, relief agencies say.

Another Houthi spokesman, Brig Gen Yahia Sarie, was quoted as saying that the missile struck and destroyed the air traffic control tower and that US-made air defence systems were unable to intercept it.

The Saudis said on Tuesday that they had intercepted two drones aimed at the south-western city of Khamis Mushait, which has an airbase.

The Yemeni government’s information minister, Muammar al-Eryani, said the targeting of Abha airport confirmed that Tehran was supplying the Houthi militia with ballistic missiles and expertise, adding that Iran’s policy of escalating the conflict was a clear challenge to those calling for calm.

He said “the Iranian project in Yemen is to use the territory to implement its subversive agenda. including threatening the energy sources and corridors of international trade”.

A limited and sporadically honoured UN ceasefire exists around the Yemeni port city of Hodeidah, but the Saudi-backed government is increasingly angry at what its regards as the one-sided approach of the UN special envoy for Yemen, Martin Griffiths.

The UN reiterated its support for Griffiths’ work this week in a UK-sponsored statement, while the Yemeni foreign minister, Khaled al-Yamani, was reported to have resigned after what was seen as a rift with Riyadh over the antagonism towards Griffiths.

The escalation between the Houthi movement and Saudi Arabia comes at a critical moment in the dispute between the US and Iran. Tehran is threatening to resume enriching uranium towards weapons-grade level on 7 July if US sanctions are not lifted or its European allies fail to offer new terms for the nuclear deal.

Before boarding his plane at Tokyo’s Haneda airport, Abe acknowledged the challenges ahead. “There are concerns over rising tension in the Middle East,” he said. “Japan wants to do as much as possible towards peace and stability in the region.”

In a sign that Abe hopes he can achieve a breakthrough in the US-Iran impasse, he spoke this week to Trump, the Israeli prime minister, Benjamin Netanyahu, and the crown prince of Saudi Arabia, , Mohammed bin Salman, and his counterpart in Abu Dhabi, Sheikh Mohammed bin Zayed Al Nahyan, all of whom are fierce critics of Tehran.

The danger is that the dispute could grow from a row over the terms of the nuclear deal into a wider struggle over Iran’s behaviour across the Middle East. The US special representative for Syria engagement, James Jeffrey, said on Tuesday that Iran’s presence in Syria was part of a “hegemonic quest to dominate the Middle East”.

Agence France-Presse contributed to this report