He is the first person executed after Prime Minister Sheikh Hasina formed a special tribunal in 2010 to try people suspected of crimes during the war. Most of the defendants are opposition members.
Mollah's party says the trials are an attempt to weaken the opposition and eliminate Islamic parties. Authorities have denied the allegation.
His execution had been placed on hold Tuesday night just before he originally was to have been put to death. The Supreme Court rejected his final appeal.
Jamaat-e-Islami, an ally of the main opposition Bangladesh Nationalist Party, had warned of "dire consequences" if he were executed.
Security was tight around the jail in Dhaka where he was hanged. Extra police were deployed along with paramilitary guards on the streets of the capital.
Earlier Thursday, party activists clashed with police, torched or smashed vehicles and exploded homemade bombs in three other major cities - Chittagong, Sylhet and Rajshahi, TV stations reported. Scores of people were injured in the latest violence to hit the South Asian country, which has seen weeks of escalating tension as it struggles to overcome extreme poverty and rancorous politics.
Security officials opened fire to disperse opposition activists in eastern Bangladesh, leaving at least three people dead and 15 others wounded, Dhaka's leading Bengali-language newspaper, Prothom Alo, reported.
The violence broke out in Laxmipur district, 95 kilometres east of Dhaka, during a nationwide opposition blockade after elite security forces raided and searched the home of an opposition leader, the report said.
The execution complicates an already critical political situation in Bangladesh, where the opposition has carried out violent protests for weeks demanding an independent caretaker government to oversee the general election set for January 5.
The government has rejected that demand and said a political government headed by Hasina will conduct the election, although the opposition alliance led by former Prime Minister Khaleda Zia plans to boycott the vote. Weeks of blockades and general strikes have left nearly 100 people dead since October.
Mollah was found guilty by the special tribunal in February of killing a student and a family of 11 and of aiding Pakistani troops in killing 369 other people during the independence war. He was sentenced to life in prison, but the Supreme Court changed that to a death sentence in September.
On Tuesday, the United Nations high commissioner for human rights, Navi Pillay, urged Hasina to stay the execution, saying the trial did not meet international standards.
Until it gained independence in 1971, Bangladesh was the eastern wing of Pakistan. Mollah's party campaigned against independence.
- AP
