The operation follows one by Shiite militant group Hezbollah last month against jihadists in another border area further south.
Hezbollah is deeply embroiled in the civil war that has raged in neighbouring Syria since 2011, fighting in support of President Bashar al-Assad's regime.
The Lebanese army has sought to keep out of the conflict but has been forced to take action since jihadists of IS and then Al-Qaeda affiliate Al-Nusra Front assaulted the border town of Arsal in 2014 and abducted 30 soldiers and police.
"In the name of Lebanon, in the name of kidnapped Lebanese soldiers, in the name of martyrs of the army, I announce that operation 'Dawn of Jurud' has started," army chief General Joseph Aoun said Saturday.
He was referring to two mountainous border areas -- Jurud Ras Baalbek and Jurud al-Qaa -- where IS has been active.
"The army is confronting the Daesh terrorists to chase them out and recover territory," army spokesman General Ali Kanso said, using an Arabic acronym for IS. "We have no fear of Daesh."
Kanso said the army believed there were around 600 IS fighters in the two areas, controlling some 120 square kilometres (46 square miles) of territory.
Hezbollah said it had launched a simultaneous operation against the jihadists from the Syrian side of the border but the army spokesman denied there had been any coordination.
Nine Lebanese soldiers captured during the 2014 raid are believed to remain in the hands of the jihadists.
Four were executed by their captors while a fifth died of his wounds. Sixteen were released in a prisoner swap in December 2015.
Last month, Hezbollah carried out a six-day offensive against IS and Al-Qaeda's former affiliate in the Jurud Arsal district further south.
It ended with a ceasefire under which nearly 8,000 refugees and jihadists were transported to a jihadist-held area of northwestern Syria in return for the release of five captured Hezbollah fighters.
The evacuations were completed on Monday.
Hezbollah chief Hassan Nasrallah vowed earlier this month to clear the whole border area of jihadists, saying it was in the interests of both Lebanon and Syria.
Lebanon, a country of some four million people, hosts more than a million Syrian refugees.
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