The attack was later claimed by the Islamic State group, which has regularly targeted Hasakeh and briefly captured parts of the city earlier this year before being expelled.
The Syrian Observatory for Human Rights said 13 civilians were killed in the two blasts, along with six members of the Kurdish security services and seven fighters from a pro-regime militia.
Syrian state television also reported the explosions, though it gave a toll of 20 dead and no breakdown of the toll.
That attack killed the six Kurdish forces along with 10 civilians, the Observatory said.
The second blast hit the Mahata district, apparently targeting a headquarters of the local National Defence Forces pro-regime militia.
Seven members of the force were killed along with three civilians, the Britain-based monitor said.
The group added that around 80 people had been wounded, and some were in a serious condition.
Control of Hasakeh city -- and other parts of the province itself -- is divided between Kurdish militia and forces loyal to President Bashar al-Assad.
The group entered the city and seized several neighbourhoods in June, but was expelled a month later after battles involving both regime troops and Kurdish fighters.
In a statement posted on social media, IS claimed responsibility for the two attacks, confirming they were suicide bomb blasts.
More than 240,000 people have been killed in Syria since the conflict began in March 2011 with anti-government protests.
The fighting has since evolved into a complex civil war involving rebels, the regime, jihadists such as IS and Kurdish fighters.
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