They hit the city from several points of ingress, attempting to overrun it and expel US -backed forces from the city.
The thousands of Western forces still there are focused on providing support and training to Afghan forces and carrying out counter-terrorism operations.
United States aircraft conducted at least four air strikes but details of the fighting were unclear as most of the city's telecoms masts were destroyed in fighting over recent days.
Residents were hiding in their homes or attempting to flee the fighting, with basic provisions becoming increasingly scarce and expensive. That said, clearing operations are ongoing and sporadic clashes are occurring.
"The Afghan National Defense and Security Forces continue to hold their ground and maintain control of all government centers", in Ghazni, he said in a statement.
"The Taliban have failed in reaching their goal", said Col.
Afghan army chief General Mohammad Sharif Yaftali told a news conference Sunday in Kabul his forces were conducting counter-offensives to evict Taliban fighters from the city.
Militants destroyed a telecommunications tower outside the city, cutting all cellphone and landline access to Ghazni.More news: NBA, USA Basketball Blindsided By NCAA's Proposed 'Elite' Prospect Rule
He said however that foreign fighters, including Pakistanis and Chechens, are involved in the battle for Ghazni.
Mustafa's wife, Razia, said they had no food, water or and electricity for four days. Most were believed to have been security forces personnel.
The city's health director told the BBC that at least 16 people had been killed and another 40 wounded.
"I went to the hospital, but it was full of dead bodies with a very bad smell", he said.
Yet any time the Taliban are invading key cities on major highways in Afghanistan, it's consequential, even if the city didn't fall in the first push. They said they "liberated" the central prison in Ghazni and freed all Taliban prisoners.
The attack on Ghazni leaves the future of peace talks with the Taleban uncertain. Mohammad said locals were coming out of Ghazni and military convoys were heading in. The government had been considering a ceasefire for the upcoming Eid al Adha holiday, similar to a truce in June over the Eid al Fitr holiday.
The meetings in Uzbekistan follow recent reports that the Taliban has also sent similar delegations to China, illustrating the group's rising ambitions to engage in independent talks with foreign governments. Islamabad denies it is helping the Taliban. The Taliban said they expect another round of talks. USA and North Atlantic Treaty Organisation forces remain in Afghanistan mainly in a supporting and training role in the 17-year-old war. Although Kabul is also supplied from Pakistan, the trade route to Iran and the Gulf supplies the northern half or the country with many essentials.
The Taliban were ousted from power in 2011 following a U.S. invasion and ongoing occupation of the country. At the height of the troop surge ordered by former U.S. President Barack Obama, thousands of North Atlantic Treaty Organisation troops were deployed in the city and province, including more than 2,500 from Poland.
Information for this article was contributed by Maamoun Youssef of The Associated Press.