Other residents, who spoke on condition of anonymity because of fears of possible Taliban retribution, said Taliban militants began their operation at around 2130 GMT on Monday (2am) attacking the city from several directions.
He said 25 government troops also died. He said fighters launched attacks from multiple directions, after which they overran several checkpoints in the city.
Reinforcements including commando units as well as the Afghan air force were battling Taliban fighters around the city, he said.
Afghan and North Atlantic Treaty Organisation warplanes also took part in the battle.
NATO Resolute Support (RS) mission has arrived in western Farah province to support the Afghan led offensive against the Taliban, a statement from the RS mission said on Tuesday.
"The situation is under control and will change by the end of the day", he said.
However the speed of the attack, which saw large parts of the city including police headquarters and the governor's compound under siege at different times, caused shock among residents. He said 10 Taliban militants were killed.
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Fighting and sporadic clashes are still ongoing at midday in Police District 3 and in western part of the city.
Interior Ministry spokesman Najib Danish said at least six security forces were killed and 12 others, including the deputy provincial police chief, were wounded in the fighting.
Another provincial council member, Dadullah Qani, confirmed Hissaini's comments.
"The people are running from the city and the situation is very bad", said a resident of Farah.
Farah is a poppy-growing province in an isolated region of Afghanistan. There are plans for a section of the multi-billion-dollar TAPI (Turkmenistan, Afghanistan, Pakistan and India) gas pipeline to traverse it.
The targeting of Farah, coinciding with gains by the Taliban since the launch of a spring offensive less than three weeks ago, forced Afganistan's President, Ashraf Ghani, to call an urgent security meeting with senior defense officials on Tuesday.
Farah, which borders Iran, has been the scene of intense fighting in recent years.
Although the insurgents have been unable to take and hold any provincial center, they are active across Afghanistan and the government has firm control over no more than 56 percent of the country, according to US estimates.