In Ghazni, the attack began around 2 am (local time) with intense gunbattles raging and fires burning in several shops in the city's residential areas, provincial police chief Farid Ahmad Mashal told The Associated Press.
After the assault, police conducted house-to-house searches to discover how the Taliban managed to infiltrate deep into the city, which is 120km south of Kabul.
"So far 140 enemy forces have been killed or wounded", the group said.
They came close to repeating the feat in 2016, and in May this year they almost overran the western city of Farah before being beaten back with the aid of USA forces.
At least one Afghan soldier has been killed and seven others wounded in the fighting, provincial governor spokesman Arif Noori said.
Mashal said 12 police had been wounded and there were more than 100 other casualties.
A second government official said it was too risky for people to leave their homes and he had no immediate details on casualties.
USA forces "responded with close-air support" in Ghazni, while Afghan security forces "held their ground" and maintained control of all government buildings.
Civilian houses and army checkpoints have come under mortar attack and the bodies of dozens of Taliban fighters are in the streets, he added.
Taliban spokesman Zabihullah Mujahid said parts of the city had been seized by the Taliban and many people were killed.
"Initial reports indicate minimal Afghan security force casualties", the U.S. spokesman told AFP, adding that American forces deployed attack helicopters and conducted a drone strike in the response.
"Another failed attempt by Taliban to seize terrain, while creating strategically inconsequential headlines", U.S. Forces-Afghanistan tweeted.
By midday Friday, O'Donnell said fighting continued inside Ghazni, prompting the USA forces to return American attack helicopters and fighter jets to the area after they were initially pulled back.
Police special forces have also been deployed to help block the Taliban advance, an Afghan security official said.
Taliban fighters armed with heavy weapons tried to overrun the provincial capital, located on the key highway between Kabul and Kandahar, last night.
The insurgents have stepped up attacks across the country since North Atlantic Treaty Organisation and the United States formally ended their combat mission in 2014, and have seized control of several districts.
An Islamic State affiliate has carried out dozens of deadly attacks in recent years, mainly targeting security forces and minority Shiites.