Bassam Ridha, an adviser to the Iraqi prime minister, confirmed at a news conference that the execution took place at 3 a.m. local time, "according to the Iraqi High Tribunal."
Government spokesman Ali al-Dabbagh said Barzan's head was severed from his body during the hanging, which Bassam Ridha called an "act of God."
Ridha said the bodies of Barzan and al-Bandar would be given to their families for burial "in the next couple of days."
A special court sentenced Barzan and al-Bandar on November 5 to death along with former President Saddam Hussein. Hussein's execution on December 30 drew criticism after unofficial mobile-phone footage was released showing him being taunted and insulted in his final moments.
Ridha said this time nothing similar happened. "No violation has been recorded whatsoever, neither any slogans or any [other] violation that we have recorded," he said.
Barzan and Bandar were ruthless enforcers of Hussein's will.
Barzan was the head of the intelligence service, or the Mukhabarat, in the 1980s. He was especially feared in Hussein's heyday. On at least one occasion, he is alleged to have calmly eaten a plateful of grapes while supervising a torture session.
Al-Bandar, the former head of Iraq's Revolutionary Court, was found guilty of sentencing 148 Shi'a to death after a 1982 assassination attempt on Hussein. He said in March 2006, during a Baghdad court hearing on the Al-Dujayl case, that he did not accept any guilt.
"Iraq then, your honor, was in a state of war," he said. "And second, they attacked the president of the republic, and they acknowledged [the attack] was prepared by Iran to overthrow the regime."
Iraqi President Jalal Talabani, an opponent of capital punishment, said last week that he thought there should be a delay in executing the two condemned men. The president left the country on January 14 to visit Syria.
Meanwhile, European Commission President Jose Manuel Barroso and Italian Prime Minister Romano Prodi condemned the execution, saying they continue to be opposed to the death penalty.