"At 10am in Baghdad the first executions were carried out since the fall of the regime, against three criminals," he said on Thursday.
Iraq's presidency had signed the death sentences for the three men found guilty by a criminal court in Wasit province in southeastern Iraq of murder, kidnapping and rape.
Thousands of Iraqis have died in violence since the 2003 US invasion, with fighters battling US troops and the US-backed government.
Crime and violence
Ordinary crime has also become rampant against a background of daily bombs, kidnappings and assassinations.
Many Western governments and rights groups had hoped the death penalty would be outlawed in Iraq after the rule of the Baath party, accused of killing hundreds of thousands of people.
"This is not an easy thing to do," Kubba said. "Despite all the condemnation from states who want us to abolish capital punishment, I think capital punishment will help us deter some criminals."
President Jalal Talabani, who opposes capital punishment, did not sign the death sentences, but his deputy signed on his behalf.