"Our jails are infiltrated by the militias from top to bottom, from Basra to Baghdad," Pusho Ibrahim Yei, an ethnic Kurd, was quoted by the US newspaper on Friday as saying.
Of special concern were the prisons run by Iraq's interior ministry that house 1,797 inmates, 90% of whom are Sunni Arabs.
Yei also said the government had suspended the transfer of prisons and prisoners from US to Iraqi control.
"We cannot control the prisons. It is as simple as that," he said.
Yei said he had written to the US officer in charge of US-run prisons in Iraq, asking him to suspend plans to transfer five facilities housing more than 15,000 inmates to Iraqi control, saying his ministry was "unprepared" for that.
US army Major-General John Gardner told The Washington Post that the transfer would not take place "until each respective facility and the Iraqi corrections system have demonstrated the ability to maintain" US standards of care and custody.
Yei said that because of concern over treatment of inmates at prisons run by the interior and defence ministries, the Iraqi police and army had agreed to turn over all their prisoners by the end of June to the justice ministry, whose facilities now house 7,426 inmates.
Some local officials, however, were already resisting the transfer order, the Post said.
Without naming Shia militias as the abusers, Yei said since 2004 they had released or helped about 725 prisoners in several cities to escape from jail.
In some instances, Sunni and foreign inmates were taken out of jail and shot to death by the militias, he added.
Salam al-Zobaie, the deputy prime minister, described the treatment of prisoners in the interior ministry prisons as "inhumane", and produced photographs showing signs of torture on prisoners' bodies.
Muhammad al-Dayni, a Sunni parliament member who made a surprise visit last week to an interior ministry prison in Baquba, north of Baghdad, said: "The detention facilities of the ministries of defence and interior are places for the most brutal human rights abuse."
Dayni said he saw as many as 120 detainees kept in a 10.5m by 12m cell. "They told us that they've been raped.
"Their families were called in and tortured to force the detainees to testify against other people," he said.
In related news, speaking to Aljazeera on Friday over phone from Baghdad, the head of the Sunni Waqf (Endowment), Shaikh Ahmed Abdal Ghafur al-Samarrae, said the body has decided to close Basra's mosques in protest against the assassination of Shaikh Yusif al-Hassan.