Students in their tens of thousands have brought parts of the capital to a standstill after two teenagers were killed by a speeding bus.
The unrest quickly spread beyond the capital. Authorities have shut down mobile internet services across swathes of the country, officials and local media said.
On Saturday the protests took a violent turn in Dhaka's Jigatala neighborhood, with more than 100 people injured as police fired rubber bullets at demonstrators.
A car carrying US ambassador Marcia Bernicat was also attacked but she escaped unscathed, the embassy said.
The violence continued Sunday with police firing tear gas into a large crowd marching toward an office of the ruling Awami League party, an AFP correspondent said.
Hasina warned that a "third party" could sabotage the protests and put the safety of demonstrators at risk. "That's why I request all guardians and parents to keep their children at home. Whatever they have done is enough," the prime minister said from her office.
Some youngsters were rushed to hospital on Saturday, allegedly being attacked by pro-government activists, witnesses said. Hasina's warning came as protesters marched towards the scene of Saturday's clashes chanting "We want justice!"
Police denied they fired rubber bullets or tear gas at the protesters. However, hospital staff said dozens of people had been injured, some seriously, and injuries were consistent with rubber bullets. The Awami League has denied allegations that its officials beat up students.
On Saturday, US ambassador Bernicat's vehicle was set upon by a mob.
"As she was leaving about 11 pm and getting into her car, a group attacked her car," said rights activist Badiul Alam Majumder, who was hosting the ambassador for dinner.
The US embassy confirmed an official vehicle "was attacked by a group of armed adult men" but the envoy and her team departed unharmed.
Newspaper headline: Dhaka paralyzed for eighth day