Taking the strategically important Ghazni increases the likelihood the Taliban could take the capital Kabul.
There is also heavy fighting in the second largest city, Kandahar.
The insurgents have moved at speed, seizing new territories almost daily, as US and other foreign troops withdraw after 20 years of military operations.
They now control a third of Afghanistan's regional cities and most of northern Afghanistan, in a crushing blow for government security forces.
Thousands of civilians have been displaced and fled to Kabul to try and escape the fighting.