The clash occurred Wednesday night at the border in southwestern Saudi Arabia's Asir region, when militants in a mountainous area on the Yemeni side fired on Saudi border troops, prompting them to return fire, SPA reported.
Besides the slain border guard -- identified as Cpl. Salman Ali Yahya al-Maliki -- 10 others suffered injuries that were not life-threatening, the Saudi media outlet said.
On Thursday, Houthi rebels seized the presidential palace in the southern Yemeni port city of Aden, a neutral security official and two Houthi commanders in Aden told CNN.
The palace had been where President Abdu Rabu Mansour Hadi, who fled the country last month, had stayed for a period of time after Houthis secured control of the capital, Sanaa, in January.
According to the security official, Houthis now control most large districts in Aden amid ongoing clashes.
Yemeni special forces, considered pro-Houthi and loyal to former President Ali Abdullah Saleh
, arrived in Aden on Thursday in an attempt "to secure the city," a Saudi official with knowledge of the situation told CNN.
"I can now confirm that it is not Saudi special forces who have landed but Yemeni special forces loyal to Saleh who landed by small boat. They are few in number and have made their way to the old parliament building," the intelligence official said.
U.S. Navy warships, meanwhile, were patrolling off Yemen in search of suspicious shipping in the region, in particular any weapons shipments from Iran to the Houthi rebels they are supporting in Yemen, a U.S. defense official told CNN.
The Navy on Wednesday came across a Panamanian-registered cargo ship that had left an Iranian port, the official said. Based on the route of the ship from Iran to Yemen, the Navy declared it a "vessel of interest" and requested permission from the captain to search it.
The ship was found to be carrying cement and construction materials, the official said. It was sent on its way.
The offshore area has been a route for weapons smuggling into Yemen, as well as Sudan and Gaza, so the Navy has maintained a regular surveillance presence there. The official said that increased fighting in the last several days in southern Yemen has resulted in other ships trying to dock to help civilians leave the country.
Marie Claire Feghali, an International Committee of the Red Cross spokeswoman in Sanaa, described events in Aden as "extremely fluid and very, very complicated."
"Hospitals are still struggling, running out of supplies. So are we," she said. "We cannot move much in Aden due to the intense fighting and chaos."
A hospital near heavy fighting was forced to shut down.