Lebanon's army began its operation in the Jurud Ras Baalbek and Jurud al-Qaa region early on Saturday, and in the first day captured around 30 square kilometres of territory, a military spokesman said.
"That is around a third of the area controlled by the terrorists," Brigadier General Nazih Jreij said late Saturday.
Jreij said 20 IS fighters had been killed in the clashes, and 10 Lebanese soldiers wounded.
Lebanon's battle against IS comes as the jihadist group faces assaults in both neighbouring Syria and Iraq, where the government early Sunday announced an offensive on the group's bastion of Tal Afar.
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Lebanese soldiers raised the Spanish flag on a hilltop captured from IS on Saturday in a tribute to the victims of those attacks, the army said.
Lebanon's military estimated around 600 IS fighters were present in the Jurud Ras Baalbek and Jurud al-Qaa areas, controlling around 120 square kilometres of territory before Saturday's advances.
On Sunday, Lebanon's National News Agency reported soldiers were firing heavy artillery and rockets at IS positions in Jurud Ras Baalbek.
The army's assault comes after Lebanon's powerful Shiite militant group Hezbollah waged its own campaign against jihadists operating in another border area south of the current battle.
The agreement saw around 8,000 refugees and jihadists transported to a jihadist-held area of northwestern Syria in return for the release of five captured Hezbollah fighters.
Hezbollah said Saturday it had launched a simultaneous operation against IS from the Syrian side of the border, where the group's fighters are battling alongside President Bashar al-Assad's army against rebels.
Lebanon's army denied any coordination with Hezbollah or the Syrian army.
Hezbollah's War Media outlet said its fighters and the Syrian army had on Saturday "managed to liberate 87 square kilometres of the total area controlled by the Daesh (IS) organisation... In western Qalamun region" of Syria.