The clashes, which erupted in the early hours of Saturday, were continuing for a second day, the Syrian Observatory for Human Rights said.
"So far, eight Kurdish People's Protection Units (YPG) fighters and security police have been killed, along with nine regime soldiers and militiamen," said Observatory director Rami Abdel Rahman.
A woman civilian was also killed in the fighting on Saturday.
The clashes broke out after Kurdish fighters detained around 10 regime loyalists they accused of seizing part of a demilitarised zone.
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Under a deal agreed last year, Kurdish forces control around 30 percent of the city's Kurdish and mixed Kurdish-Arab districts, with regime forces controlling most of the city's majority-Arab neighbourhoods.
Certain districts are off-limits to both sides under the deal.
The two sides have fought together to keep Islamic State group jihadists out of Hasakeh, a provincial capital of some 200,000 people, but Kurdish relations with government forces are complicated.
The regime withdrew from most Kurdish-majority areas of Syria in 2012, focusing its forces on fighting the burgeoning Sunni Arab-led rebellion.
Since then, the Kurds have worked to build autonomous local governments in the three regions where they form the majority population.
Kurdish forces have taken over most security responsibilities in those areas.