The Houthi-run Masirah TV said the rebels had launched a cruise missile attack on Abha airport, which is about 125 miles north of the border with Yemen and serves domestic and regional routes.
A Houthi military spokesman on Tuesday threatened that the group, which has been battling the Saudi military alliance in Yemen for four years, would target every airport in Saudi Arabia and that the coming days would reveal “big surprises”.
Saudi Arabia and the UAE intervened in 2015 to support the Yemeni government against Houthi rebels, in a war that has left around 10,000 people dead.
The Saudi-led military coalition said, which has been fighting the Iran-aligned Houthis in Yemen, said it would make a “firm” response.
A coalition statement, carried on Saudi-owned Al Arabiya TV, said a projectile hit the arrivals hall at Abha airport, causing material damage.
Three women and two children were among the wounded, who were of Saudi, Yemeni and Indian nationalities, it said.
“Targeting Abha airport proves that the Houthis have obtained advanced weapons from Iran,” the coalition statement said, saying the attack could amount to a war crime.
The attack comes amid tensions in the Gulf, which has seen a massive US military build up in response to what it says are Iranian threats.
In May a pipeline in Saudi Arabia came under attack from drones armed with explosives flown by the rebels.
The Houthis have repeatedly launched drone and missile attacks on Saudi cities, but two Saudi sources said this was the first time an Aramco facility was hit by drones.
According to experts, the attack represented a massive leap in the Houthi’s military capability.
The US also has accused Iran of being behind a string of incidents, including alleged sabotage of oil tankers off the coast of the United Arab Emirates and a rocket that landed near the US Embassy in Baghdad.