The decision was taken weeks after schools were closed in mid-March to combat the spread of the virus, leaving many students and teachers to adapt to distance learning.
"All primary and secondary school students will move on to the next class," the government announcement said, according to state news agency SANA.
Brevet and baccalaureate examinations -- usually taken at the end of secondary school and high school respectively -- will still be sat by 557,000 students, according to the education ministry.
The government will increase the number of exam centres to ensure "distance" between students, SANA reported.
After schools were shuttered, some institutions moved to online teaching, while a specialised education ministry TV channel broadcast Arabic, English, mathematics and science courses.
But daily power cuts that can last for hours and capped, costly household internet have posed challenges to distance learning efforts in the country wracked by war since 2011.
Universities will remain closed at least through the end of the holy month of Ramadan in late May, according to SANA.
Damascus has officially reported 42 cases of COVID-19 and three deaths from the disease in government-controlled territory.
Authorities have adopted a series of measures to stem the spread of the virus, closing shops and restaurants as well as imposing a strict curfew and movement restrictions.