"Armed men infiltrated at Jabal al-Dukhan... and opened fire on the border patrol with weapons of different calibre," SPA reported citing an unidentified government spokesman.
The Houthi fighters in a statement on their website said they seized the territory inside Saudi Arabia where the battle took place.
"Complete control has been taken yesterday evening of Jabal al-Dukhan ... after attacks during which the aggressor was eliminated," the fighters said .
The Houthis accused Riyadh of permitting Yemeni government troops to launch attacks against them from a Saudi security installation in Jabal al-Dukhan.
Rebellion
The conflict between the Houthi fighters and the Yemeni government began in 2004, but intensified last August when fighters stepped up the pressure on government forces.
Sanaa has vowed to crush the Houthis and launched a military offensive against them codenamed Operation Scorched Earth.
The Houthis accuse the government of marginalising them economically and politically and are seeking greater autonomy for themselves.
They are known as Houthis after their late leader, Hussein Badr Eddin al-Houthi, a Zaidi leader who was killed by the Yemeni army in September 2004.
An offshoot of Shia Islam, the Zaidis are a minority in mainly Sunni Yemen but form the majority in the north.
Thousands of people have died and at least tens of thousands have been displaced due to the conflict, according to the Red Cross and UN.