The town in Syria's Aleppo province is the last stronghold of the jihadist group in the region, and has also been the target of an assault by Syrian government forces.
The Syrian opposition meanwhile announced the formation of a delegation to attend a new round of UN-sponsored peace talks in Geneva on February 20.
IS has come under pressure from simultaneous offensives in both Syria and Iraq, where the group seized large swathes of territory in 2014 and proclaimed an Islamic "caliphate".
"Our forces entered the centre," he added, saying it was "only a matter of time" before the alliance of Turkish forces and rebels took control of the town.
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"Daesh forces have begun leaving Al-Bab completely," he said, using an Arabic acronym for IS.
Turkish forces and allied rebels entered Al-Bab for the first time on Saturday, from the west, according to the Syrian Observatory for Human Rights.
The Britain-based monitor reported heavy clashes inside western Al-Bab on Sunday, as well as on the northern edge of the town, where Turkish forces and rebels were advancing but had not yet entered the town.
That raised to 67 the number of Turkish soldiers killed since Ankara began its Operation Euphrates Shield in August, targeting both IS and the Kurdish YPG militia.
Al-Bab has been a key target for both Operation Euphrates Shield, but also Syrian government forces, and Ankara now finds itself effectively jointly besieging the town with President Bashar al-Assad's forces despite opposing his government.
Syria's army has advanced towards Al-Bab from the south, and on Monday severed the last road leading into the town, completing its encirclement.
More than 310,000 people have been killed in Syria since the conflict began in March 2011 with anti-government protests, and successive rounds of peace talks have failed to produce a political solution.
Invitations to new talks on February 20 in Geneva have been delayed while the opposition forms its delegation.
Today, the leading components of the political opposition announced a 21-member delegation, with a new delegation head and chief negotiator.
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