The first bomb hit Kurdish forces in the Khashman district, the Syrian Observatory for Human Rights (SOHR), a UK-based activist group, said.
The second targeted pro-regime forces in the city centre, the SOHR said.
The city is largely held by Kurdish fighters, with some parts held by the government, but Islamic State militants have also made efforts to capture it.
State TV showed footage of buildings destroyed by the attacks and said rescue teams were searching the rubble of collapsed buildings for more casualties.
In June, the United Nations said an estimated 50,000 people had been displaced within Hassakeh after an IS offensive there. Another 10,000 fled north towards the Turkish border.
In four years of armed conflict, more than 200,000 Syrians have lost their lives and 11 million have been forced from their homes.
Two car bombs in the Syrian city of Hassakeh have killed at least 20 people and injured 40, state TV reports.