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 on Thursday.
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 kilometers (60 miles) 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 Jan. 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.
Also Thursday, security officials detained former President and military dictator H.M. Ershad nearly two hours after Mollah's execution.
Ershad, who ruled the country for nearly nine years until 1990, when he was overthrown in a mass movement led by Hasina and Zia.
He was brought to a military headquaters in Dhaka by security forces.