"Both of them were hanged simultaneously on two separate platforms," the police official said.
Islamist opposition leader Ali Ahsan Mohammad Mujahid and Salauddin Quader Chowdhury, former legislator from former premier Khaleda Zia's Bangladesh Nationalist Party (BNP), were hanged shortly after President Abdul Hamid rejected their appeals late on Saturday for clemency.
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Mujahid, 67, of the Jamaat-e-Islami party, and Chowdhury, 66, were hanged at Dhaka Central Jail. The Supreme Court had previously rejected their appeals against a death sentence imposed by a special tribunal for genocide and torture of civilians during the conflict.
The Border Guard Bangladesh paramilitary force has been deployed across the country to tighten security.
Muslim-majority Bangladesh, until 1971 East Pakistan, has seen a rise in Islamist violence in recent months, with two foreigners and four secular writers and a publisher killed this year.
Mujahid was found guilty on five charges including torture and the murders of intellectuals and minority Hindus while he commanded Al Badr, an auxiliary force of the Pakistani army, during the war to break away from Pakistan.
Chowdhury, former legislator from former premier Khaleda Zia's Bangladesh Nationalist Party, was convicted in October 2013 on charges of genocide, religious persecution, abduction and torture during the war.
"While we are saddened that we have lost our father by way of a motivated and predetermined trial and where the country is gagged from speaking out, we find hope in the fact that the international community recognises the injustice and that fairness and truth shall be restored in Bangladesh," Humam Quader Chowdhury, a son of Chowdhury, told Reuters.
"We fought for them under the law and we have been defeated in the legal fight," defence councillor Khandker Mahbub Hossain told Reuters.
Prime Minister Sheikh Hasina opened an inquiry into crimes committed during the war in 2010, paving the way for prosecutions by a war crimes tribunal that Islamists have denounced as part of a politically motivated campaign aimed at weakening Jamaat-e-Islami's leadership.
Two Jamaat leaders have been executed, one in December 2013 and another in April. They declined to seek clemency from the president.
BNP spokesman Asaduzzaman Ripon said: ”Salauddin has fallen victim to persecution because of his political identity, and he has been denied justice.”
Moqbul Ahmed, acting Amir of Jamaat, said in a statement that Mujahid was a victim of government conspiracy. He called a day long general strike on Monday across the country.
The government denies accusations of interference in the judiciary.
East Pakistan broke away to become independent Bangladesh after a war between India and Pakistan. About three million people were killed.
