Hamad al-Kaabi was killed after heavy gunfire hit a border post in the southwestern region of Jazan, said the statement published by the official SPA news agency.
Saudi Arabia is leading an Arab coalition that has been carrying out air strikes against Iran-backed rebels across Yemen since March 26.
More than 50 people, most of them soldiers, have been killed in shelling and skirmishes on the Saudi side of the border with Yemen since the coalition's campaign began.
In Yemen, the United Nations says the war has killed nearly 4,500 people, including many civilians.