At least 10 insurgents and two members of the security forces had been killed so far, Defense Ministry spokesman Mohammad Radmnish said.
No one immediately claimed responsibility for the attack. The fighters launched the attacks from four directions on the city, which has over 50,000 residents.
Local military experts believed the Taliban offensive followed their loss of control over the southern province of Helmand.
In January, NATO's Resolute Support Mission, tasked with building up the Afghan government's security forces, assessed that almost 15 percent of the country's 407 districts - more than 20 percent of the landmass - was under insurgent control or influence, according to a watchdog report released this month.
The assault comes as the Taliban are stepping up their spring offensive, in an apparent rejection of a peace talks overture by the Afghan government.
Another resident said: "The government forces have conducted a number of operations in different parts of the city in the past few days, but this hasn't had any results".
On Monday, CEO of the National Unity Government (NUG) Abdullah Abdullah said soldiers have been under siege for months, but relevant security officials are busy with other issues.
Clashes between militants and security forces forced Farah residents to close shops and businesses.
The latest reports say the terrorist group is still two-miles away from the city center. and the battle is still underway.
Most people are staying indoors. "Gunfight, airstrikes and explosions are taking place around the city", he said.
Many radio and television channels in the province have stopped broadcasting, fearing for their employees' lives, according to media watchdog Nai.
Poor coordination among security forces and the Taliban's access to more sophisticated weapons have been cited as major factors in the militants' advance.
Farah is a poppy-growing province in an isolated region of Afghanistan. During 2017, they failed three times to seize the city.
The attack on Farah is the first siege of a provincial capital this year.