The triple attack took place in Qara Tappah in ethnically mixed Diyala province, an official from the Kurdish Asayish security forces said.
He said the first bomber detonated an explosives vest at the gateway to a security compound that also houses the office of a main Kurdish political party. Minutes later, two suicide bombers plowed cars filled with explosives into the compound, causing heavy damage, he said.
The authenticity of the online statement could not be independently verified, but it was posted on a Twitter account frequently used by the militant group.
The group has seized towns in volatile Diyala province and has clashed with Kurdish forces there.
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Hospital officials confirmed the casualties. All officials spoke on condition of anonymity because they were not authorized to speak to the media.
Bashir al-Dalawi, a member of the Qara Tappah municipal council, where the attacks took place, said that the bombs detonated near a government building, leading to the higher death toll.
In the Anbar attack, Brig. Gen. Ahmed al-Dulaimi was killed while traveling in a convoy north of the provincial capital Ramadi through an area cleared by Iraqi security forces a day earlier, Anbar councilman Faleh al-Issawi said. It was not immediately clear if others were killed or wounded.
The Islamic State group and allied Sunni militants seized the Anbar city of Fallujah, parts of Ramadi and large rural areas of Anbar early this year.
Iraq's Interior Ministry confirmed al-Dulaimi's death, calling him a "hero who set a good example for self-sacrifice."
It praised his role in reorganizing the provincial police force and leading major fighting that caused heavy casualties among the militants.