"Eight people were wounded and a civilian aircraft was damaged, according to initial information," it added.
The coalition said the second attack on the airport "constitutes a war crime" after it intercepted a booby-trapped drone earlier in the day.
In a second statement, the coalition said that those injured included one Saudi national, a Nepalese, three people from India and three from Bangladesh -- one of whom was in critical condition.
The coalition also said it launched a military operation targeting a launchpad used in Yemen's capital Sanaa -- under rebel control since 2014 -- that was used to carry out the Abha attack.
It added that it "destroyed the terrorist elements responsible for the attack."
According to Al-Ekhbariya, after the coalition intercepted the first attack, shrapnel hit parts of the airport near the runway.
It added that flights had been temporarily halted "to ensure the safety of incoming and departing aircraft, as well as civilians at the airport".
The rebels have yet to comment on the incident.
Saudi Arabia intervened in the Yemen war on behalf of the internationally recognised government in 2015, shortly after the Huthis seized Sanaa.
The Iran-allied insurgents have repeatedly targeted the kingdom in cross-border attacks.
In August, the rebels escalated those operations, using drones and missiles.
Yemen's grinding conflict has claimed tens of thousands of lives and displaced millions, resulting in what the United Nations calls the world's worst humanitarian crisis.
While the UN is pushing for an end to the war, the Huthis have demanded the re-opening of Sanaa airport, closed under a Saudi blockade since 2016, before any ceasefire or negotiations.
The incoming UN special envoy for Yemen, Hans Grundberg, will officially assume his duties on 5 September.