At least 34 people have been wounded, the head of the local hospital told the BBC. No deaths have yet been officially confirmed.
Taliban insurgents told news agencies they were behind the attack.
Government offices are frequently targeted by the Taliban. Farah is a strategic province bordering Iran.
A bomb and gun attack on the governor's compound, last May, left six policemen, a civilian and four attackers dead.
Eyewitnesses, including doctors in the city, said both sides were using machine-guns and grenades.
Mohammad Yunos, Farah's deputy governor, told a private Afghan TV channel by phone that militants had set off a "large explosive-laden vehicle at the Western part of the governor's office".
The resulting blast damaged high-rise buildings, including the governor's office, other government and non-government agencies, and two banks.
After the initial assault, he told Channel One TV, that gunmen took up position in "multi-story buildings".
"Currently the gunfire is getting less intense compared to the beginning when there was gunfire and explosions," he said.
Speaking to the BBC, a senior Afghan security source said, "First there was a suicide attack, later on a group of attackers started fighting".
He also confirmed there had been fighting close to a court and a bank, both of which had been "packed with civilians".
Local shopkeeper, Sayed Jan, told the BBC, "I heard a huge bang and when I looked I saw several attackers armed with light and heavy weapons running towards the bank and the court. I left my shop.
"There is blood and broken glass. There is a heavy exchange of gunfire and my shop has become a center of the fighting. "
Taliban spokesman Qari Yusuf Ahmadi said the group had carried out a suicide attack on the governor's office at approximately 09:00 local time (04:30 GMT).
Following the blast, gunmen entered several "government institutions", he was quoted as saying by the Pakistan-based Afghan Islamic Press news agency.
