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."
Ali Abdullah Saleh, the Yemeni president, said that the military would not ease the offensive against the Houthis "until we bring this tyrannical, traitorous and mercenary group to an end".
|The Houthis have released video footage to prove their claim of Saudi cross-border raids
"The war just started two days ago, and what has been happening in the past six years was just ... training for the army units," he said.
"The war will never stop no matter how much money or martyrs it costs."
Khalid al-Dakhil, a political analyst at the King Saud University, said that it was impossible for Riyadh to ignore the conflict between the Houthis and the Yemeni government.
"The Houthis seem to be very determined in pinning down the Yemeni army, and the Saudi government cannot afford to just sit by and watch what is happening. They have to support the Yemeni government", he told Al Jazeera.
"The co-operation between Yemen and Saudi Arabia runs for a long time now, even before the Houthis.
"I don't think the Houthis have any chance of succeeding in the end. They will be isolated, they will be squeezed between two armies and I think eventually they will run out of any luck."
Hundreds of people have died in northern Yemen since the country's army began an offensive against the Houthis on August 11.
The fighters, concentrated mainly in the Saada and Amran provinces, 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 a predominantly Sunni Arabian peninsula but form the majority in northern Yemen. Only a small minority of Zaidis are involved in the Houthi uprising.
The Yemeni government accuses the Houthis of seeking to restore an imamate overthrown in a 1962 coup that sparked eight years of civil war.
The Houthis, now led by Abdul-Malik al-Houthi, Hussein's brother, insist they are fighting to defend their community against government aggression and marginalisation.