In its statement, al-Qaida in Iraq said the prison operation involved 12 car bombs, military-style barrages of rockets and mortar shells, suicide bombers and help from prisoners who had managed to obtain weapons on the inside.
Iraqi officials have said at least 25 members of the Iraqi security forces were killed in the attacks, along with at least 21 prisoners and 10 militants.
Al-Qaida boasted that its men killed more than 120 government forces, and it claimed that on its side, only the suicide bombers died in clashes that raged for hours.
Frank Finver, a spokesman for the U.S. Embassy in Baghdad, said the U.S. is "deeply concerned" by the attacks and the overall levels of violence in Iraq.
He said the U.S. is in contact with the government of Iraq to help improve its ability to weaken or defeat al-Qaida inside Iraq, but he gave no specific details on counter-terrorism cooperation.
Iraq's central government has not provided a clear account of what happened or said how many prisoners escaped Sunday night.
The Interior Ministry has said several prisoners broke out from Abu Ghraib, the infamous prison in Baghdad's western suburbs, which was the site of well-publicized prisoner abuse at the hands of the U.S. military after the 2003 invasion.
But several Iraqi officials, including members of parliament's security and defense committee, have said more than 500 inmates escaped. Both prisons house thousands of inmates, including convicted al-Qaida militants, though it appears no one broke out of the Taji prison.
Al-Qaida said in its statement that the attack freed hundreds of detainees, including more than 500 mujahedeen or holy warriors.
The authenticity of the statement could not be independently confirmed. It was posted on a website commonly used by jihadists and its style was consistent with earlier al-Qaida statements.
The prison break could help bolster al-Qaida's ranks and give them a propaganda boost.
"Al-Qaida needs as much resources as it can get, including manpower," said Paul Floyd, who served several tours in Iraq with the U.S. Army and is now a military analyst at global intelligence company Stratfor. "The larger the operation, the more successful it is, the more likely they are to claim it."
Iraqi security forces have struggled to quell the violence, angering many Iraqis. The prison attacks have only increased the sense of insecurity.