Hasina's party is blaming the main opposition Bangladesh Nationalist Party, headed by former Prime Minister Khaleda Zia and its main ally Jamaat-e-Islami, for using the sentiment of young students to create chaos for political gains.
The proposed law change would be considered by Cabinet on Monday, he said.
The violence continued on Sunday with police firing tear gas into a large crowd marching toward an office of the ruling Awami League party, an AFP correspondent said.
During the protests, school children took to the street to regulate traffic and check drivers' licenses and vehicles fitness, forcing authorities to declare school holidays. They're calling for improvements to road safety in Bangladesh, where around 12,000 people are killed in road accidents each year, according to the Associated Press.
"We have assured them that all their demands will be fulfilled and a law will be proposed in the next session of parliament", Interior Minister Asaduzzaman Khan told Reuters. There were no injuries but two vehicles were damaged.
He said: "We strongly support the spirits of the young students in Bangladesh, who showcased few examples of detecting the irregularities related to road safety violations". Ambassador to Bangladesh Marcia Bernicat was returning from dinner at the time of the attack.
In some places, there have been clashes between the protesters and police.
Last week police beat some students in their efforts to disperse them.
The protesters were attacked by some people who were alleged to be linked to the ruling party activists with some of them armed with machetes, Dhaka's Dhanmondi neighborhoods.
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"Then we broke the locks of a building and around 50 boys and girls took shelter there".
"We express our appreciation to the Dhaka Metropolitan Police and security officials for their quick and professional response following the incident", the Embassy said in a statement.
Authorities have pleaded with students to call off the protests that have almost paralyzed Dhaka and spread beyond the capital.
"There is no justification whatsoever for detaining anyone for exclusively peacefully expressing their views", Omar Waraich, Amnesty's deputy South Asia director, said in a statement.
The United Nations said it was anxious for the safety of the children and young people caught up in the protests.
Students have brought traffic to a standstill in the city, demanding justice for the two victims.
The protests spreading across Bangladesh have highlighted traffic risks in the densely populated country, where more than 4,000 people die in road accidents each year, one of the world's highest rates, the World Bank says.
No details about the number of people injured in the clash have been given. The owners and workers of the bus companies have said they will not run their vehicles unless they feel safe after dozens of vehicles were either vandalized or torched in Dhaka and elsewhere. "The commission based system must be eliminated", said Mr Shafi, who was injured while protesting over the weekend.