The offensive began over the weekend in Nangarhar province, where the IS group has had a presence for the past year, said Dawlat Waziri, spokesman for the Defense Ministry.
The offensive, part of the government's Operation Shafaq or Dawn in Pahsto started hours after an IS suicide bomber killed at least 80 people who were taking part in a peaceful demonstration in Kabul on Saturday.
President Ashraf Ghani gave the orders for the offensive, which Waziri said will consist of airstrikes and ground attacks, including those by special forces.
Clean-up operations aimed at ensuring that IS loyalists do not return to the area would follow, he added.
The offensive marks a new chapter in Afghanistan's war against insurgents. Until now they have been a largely defensive force, and have struggled to take the lead on the battlefield since the withdrawal in 2014 of most international combat forces.
The head of US and NATO forces in Afghanistan, Gen John Nicholson, has pledged to transform the Afghan military into an offensive force so they can take the fight to the insurgents, and reclaim battlefield initiative.
President Barack Obama recently expanded the conditions for US troops in Afghanistan to support Afghan offensive operations.
Nicholson can make the strategic shift from using airpower only to defend U.S. And NATO positions, to striking in support of Afghan offensives.
The US military spokesman in Kabul, Brig Gen Charles Cleveland, said American forces have been involved in the current operation in the east and are conducting "multiple counter-terrorism strikes.
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