The UN Security Council has backed Lebanon's military action against armed groups but urged it to stay out of the conflict in neighbouring Syria, as Beirut vowed no leniency for the "terrorist killers".
Twenty-two Lebanese soldiers were missing, possibly taken hostage, the army announced, and a military source said 16 others had been killed since the clashes with rebel fighters erupted on Saturday near the Syrian border.
The Security Council on Monday called on Lebanese politicians to "preserve national unity" and "refrain from any involvement in the Syrian crisis".
In-depth coverage of escalating violence across Syria
The 15-member Council "expressed support for the Lebanese Armed Forces and the Internal Security Forces in their fight against terrorism".
Analysts said the violence could be contained in the short term, but warned an aggressive military response could stoke tensions and worsen the clashes, which have also killed three civilians.
On Monday, the army deployed reinforcements and fired mortar rounds at Syrian-held positions in the mountains around the eastern border town of Arsal, saying it was advancing.
Residents fled en masse after the fighting continued overnight.
Tammam Salam, Lebanon's prime minister, pledged there would be "no leniency towards the terrorist killers and no appeasement for those who violate Lebanon's territory and harm its people".
In a statement after a cabinet meeting, he also urged France to speed up delivery of weapons for the Lebanese army being purchased under a $3bn deal financed by Saudi Arabia.
The violence in the eastern Lebanese region began on Saturday afternoon, after soldiers detained a Syrian man, Imad Ahmed Jumaa, who the army said confessed to belonging to al-Qaeda's Syria affiliate, the Nusra Front.
Police station stormed
Rebels angered by the arrest opened fire on army checkpoints and stormed a police station, killing two civilians and capturing several police.
A third civilian was shot dead by a sniper on Sunday.
The clashes are the latest evidence of how Syria's conflict, which began with protests against President Bashar al-Assad in March 2011, has afflicted Lebanon.
Despite officially distancing itself from the war, Lebanon's existing sectarian and political tensions have been worsened by the conflict next door.
It is also hosting more than one million Syrian refugees.
Hezbollah, the powerful Lebanese Shia group, has openly intervened in the Syrian conflict, dispatching fighters to bolster regime troops against the Sunni-dominated uprising.
Arsal, a majority Sunni area, is broadly sympathetic to the uprising against Assad's regime and has been regularly bombed by Syrian regime forces who say that they are targeting opposition fighters holed up in the area.