The Syrian Observatory for Human Rights said the fighting broke out at around 2 am at several points in the city, control of which is split between the two sides.
Rami Abdel Rahman, director of the Britain-based group, said there had been heavy fighting since then.
Four regime forces and a Kurdish fighter had been killed so far, as well as a woman in regime shelling of an area controlled by the Kurdish People's Protection Units.
The clashes began after Kurdish fighters detained around 10 regime forces they accused of seizing part of a demilitarised zone.
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 districts.
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Certain districts remain off-limits to both sides under the deal.
Against that backdrop, the two sides have fought jointly to push Islamic State group jihadists out of Hasakeh, but Kurdish relations with government forces are complicated.
The regime withdrew its forces from much of the country's Kurdish region in the north and northeast in 2012, focusing its attention on the burgeoning rebellion.
Since then, the Kurds have largely assumed security responsibility there and have worked to build autonomous local governments.
Hasakeh, a city of around 200,000, has remained relatively safe during the conflict, despite efforts by IS to seize outlying areas.
More than 200,000 people have been killed in Syria since the beginning of the uprising in 2011.