They said the violence included three suicide bombings followed by fighting in a market, adding that all eight attackers had now been killed. The dead include Ahmed Omed Khpulwak.
The Taliban say they carried out the attack, which comes amid renewed violence in Afghanistan. Nato says it is providing air support to Afghan forces in Tarin Kowt.
Afghan intelligence officials said at least one bomb exploded near the governor's office and one near the offices of a security firm owned by a local militia commander. It is not clear where the third bomb was detonated.
Most of the fighting took place near these offices, which are close to the main market and a building which houses a local radio and TV station.
The BBC's Bilal Sarwary says the market was attacked from four sides, but the siege was broken by elite forces.
Residents said heavy machine guns, rocket-propelled grenades and assault rifles were used by both sides.
Health officials said 22 people had been killed including three women and 40 injured most of them civilians.
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