The firefight came a week after Yemen's ex-president Ali Abdullah Saleh, who is allied with Shiite Huthi rebels, called for "revenge" following air strikes on a funeral ceremony in Yemen's rebel-held capital killed more than 140 people.
Rebels blamed the Saudi-led coalition for the October 8 strikes in Sanaa.
The Saudi interior ministry said a border post in the kingdom's Jazan region came under fire from the Huthis on Thursday night, killing a member of the Border Guards force.
At least 110 Saudi soldiers and civilians have been killed along the border, either in rebel rocket fire or armed clashes, since March last year when the coalition began an air war against the rebels.
The coalition intervened to support Yemen's internationally-recognised government after the Huthis overran much of the country.
Rebels twice this week fired ballistic missiles towards southwestern Saudi cities that host air force bases.
The missiles were shot down, according to the coalition.
In Yemen, the United Nations says 6,885 civilians have been killed, about half of them civilians.