The government spokesman said that al-Tikriti and Bandar were treated with dignity before the execution and those who attended were asked to sign a document saying they would not taunt or insult the men during the event.
"Their rights were not violated. There was no chanting," he said.
Al-Tikriti's head was detatched from his body during the hanging, al-Dabbagh added. Another official called it was an "act of God".
The bodies have been turned over to police and family members have been asked to retrieve them for burial, a source told the AFP news agency.
The United Nations had urged Iraq not to go ahead with the executions.
The two men were sentenced to death for crimes against humanity on November 5 for their part in the killings of the Shia men and youths in 1982 in retaliation for an attempted assassination of Saddam in the town of Dujail, north of Baghdad.
They had been due to be executed within a month of losing their appeal on December 26.
The pair were to have been hanged along with Saddam on December 30, but Iraqi authorities decided to execute Saddam alone on what Mowaffak al-Rubaie, Iraq's National Security adviser, called a "special day".
The executions reportedly occurred in the same Saddam-era military intelligence headquarters building in north Baghdad where the former leader was hanged two days before the end of 2006, an Iraqi general, who declined to be named, told AP.
Talabani had called for a delay in the executions last Wednesday, saying: "In my opinion we should wait ... We should examine the situation."
Talabani - who for two-decades helped lead a Kurdish revolt against Saddam - has few real powers under the Iraqi constitution and had no powers to reduce or block the sentence against the two men.
Images of Saddam's execution showed an unruly scene that brought worldwide criticism of the Iraqi government.
Video of the execution, recorded on a cell phone camera, showed the former dictator being taunted on the gallows shortly before his death.
The footage angered many in Iraq's Sunni Arab minority, embarrassed the Shia-led government and the US administration and also raised sectarian tensions.
Barzan, one of Saddam's three half-brothers, was the former head of the Mukhabarat intelligence service during the 1980s and one of the most feared men in Iraq.
A witness at his trial said Barzan had personally supervised his torture with electric shocks in Baghdad and had eaten grapes while the man screamed in agony.
Another witness described how Barzan beat her and broke her ribs after she was hung naked from the ceiling by her feet.
Bandar presided over the Revolutionary Court which sentenced the 148 Shia men and youths to death in 1982.
Along, with Saddam, they were convicted of crimes against humanity by the US-sponsored High Tribunal in November.