Saudi Arabia began air raids and artillery bombardments against the Houthi group after its fighters crossed from northern Yemen and reportedly took control of the area called Jebel al-Dukhan.
"The situation is calm ... especially in Jebel al-Dukhan, of which full control has been regained," Prince Khaled bin Sultan bin Abdul-Aziz, the assistant minister for defence and aviation, said on Saturday, according to the Saudi Press Agency (SPA).
However, there were reports that air raids along the border were continuing on Sunday.
Theodore Karasik, an analyst at the Institute for Near East and Gulf Military Analysis, said that it was likely that the apparent success of the Saudi action was only a "temporary reprieve".
"The Saudis have been able to push back the rebels but they are going to continue to have problems with the rebels if the Yemeni army is indeed using the southern part of Saudi Arabia for operations and for supply lines," he told Al Jazeera from Dubai.
"The government in Riyadh is working closely, or wants to work closely with the the government in Saana to arrest this problem.
"This draws them closer together, but from the point of view of the rebels, and also from al-Qaeda, it means they become apostates and they will go after them more."
Prince Khaled was reported as saying that three members of the Saudi security forces had been killed and 15 wounded in the fighting along the Saudi-Yemeni border.
He also acknowledged that four Saudi soldiers were missing following the five days of fighting, but dismissed claims that anyone had been captured.
Mohammed Abdel-Salam, a spokesman for the Houthis, told Al Jazeera on Friday that the men were seized after Saudi ground forces crossed into Yemeni territory.
The group has tried to prove that Saudi troops had crossed over the border by releasing video footage purportedly showing the Saudi military in Yemeni territory.
But Prince Khaled said that Saudi Arabia "has not, and will not interfere inside Yemeni borders".
Hussein Shobokshi, a columnist for Asharq Al Awsat newspaper in Saudi Arabia, said that the conflict along the border has been escalating for some time.
"We have seen a lot of drug-trafficking taking place, human-trafficking taking place, al-Qaeda members infiltrating the borders, an assassination attempt on the deputy interior minister - all these came through Yemen," he told Al Jazeera from Jeddah.
"The Saudis had to take a stand, had to take a very aggressive stand, since the Houthi aggression is not a very benign one, it is not a Yemeni one any more."