Prime Minister Begum Khalida Zia has said she will not quit.
Streets were bare and shops, schools and businesses shut after the Awami League called the strike to topple Khalida, and to protest what it called "barbaric police actions" during the stoppages.
Authorities deployed hundreds of extra police and paramilitary troops in Dhaka, the capital, and in the port city of Chittagong to try to avert violence.
"Police stormed the Awami League central office in Dhaka this morning and beat and detained our people in there," said Asaduzzaman Noor, the party's publicity secretary. "We strongly denounce such excesses."
Nur, who is also a member of parliament, added, "Police are behaving rudely and in the same pattern ... we will broaden our protest, which may not be confined to strikes only."
Business leaders say each day of production lost to strikes - called frequently by the opposition - costs the impoverished country at least $60 million.
Political observers say the country is heading into a period of confrontation after a period of relative calm after Khalida's victory over former prime minister Shaikha Hasina, leader of the Awami League, in October 2001 elections.
Nearly 20 people were injured in clashes on Sunday night between activists of the Awami League and Khalida's Bangladesh Nationalist Party in the southeastern resort of Cox's Bazar, local reporters and officials said.
"Police are behaving rudely"
publicity secretary, Awami League
About 30 shops were set afire and several vehicles damaged in the violence, they added. Police said similar violence was also reported from Bogra in the north, but gave no details.
The Awami League had already enforced day-long countrywide strikes on Thursday and Saturday last week and said Monday's strike was to protest against a police attack on a leader of its youth group.
Awami General Secretary Abd Al-Jalil said Monday's strike was called by the Jubo League after its chairman, Jahangir Kabir Nanak, was injured by police on Saturday.
"The repeated strikes called by Awami League have been designed to harm the country and its people," Khalida said. "I would like to say categorically that any trouble-mongers will be punished."
Hasina, who accused Khalida's two-year-old government of pushing Bangladesh into unprecedented lawlessness, has vowed to take harsher action unless the government behaved "politically".