In one of the blasts, a suicide bomber killed at least 13 people and wounded 18 in a crowded restaurant, killing police and civilians.
The bloodshed is a poignant reminder of the task that Mr Maliki faces in reining in the continuing unrest that has pushed Iraq to the brink of sectarian civil war.
The day before, Mr Maliki had formed a cabinet of Shi'ites, minority Sunni Arabs and Kurds to ease violence and consolidate a US-piloted transition to democracy.
US President George W Bush said the new government marks a "new day for the millions of Iraqis who want to live in freedom."
US Secretary of State Condoleezza Rice, speaking on Fox News Sunday, said it is too early to make commitments on sending home some of the 130,000 US troops.
Insurgency crackdown
Mr Maliki said his government would offer dialogue to insurgents who lay down weapons and vowed to reimpose the state's monopoly on the armed forces, cracking down on militias.
"Militias, death squads, terrorism, killings and assassinations are not normal and we should put an end to the militias," the tough-talking Shi'ite Islamist said.
Prime Minister Maliki also said a special security force would be created to help protect the capital, Baghdad.
But the prime minister stressed that the 37-member cabinet finally approved by MPs on Saturday after more than five months of coalition talks, would need to do more than resort to strong-arm tactics if it were to rein in the insurgency raging in Sunni areas.
"We have to make extraordinary efforts because we suffer from terrorism and security problems and we have administrative and financial corruption," Mr Maliki said.
Mr Maliki is trying to forge consensus among Iraq's rival communities but partisan wrangling over jobs, differences over the role of Islam and the sharing of Iraq's natural resources continues to hold-up progress.
Disputes over who would lead the key interior and defence ministries, in charge of police and the army, meant those two posts would be left vacant for now. The Iraqi prime minister said he hopes to fill the posts in two to three days.
Neighbours welcome news
Iraq's neighbours have hailed the formation of the new government in Baghdad after months of stalled negotiations.
Iran has said it was determined to improve ties with its former enemy.
"Iran welcomes the formation of the new Iraqi government and we are determined to expand and solidify our brotherly relations with Iraq," foreign ministry spokesman Hamid Reza Asefi told reporters in Tehran.
Jordan's King Abdullah II telephoned Iraqi President Jalal Talabani on Saturday to congratulate him.
The king called the new government "a significant step towards building a new Iraq that would be able to fulfill the aspirations of its people for a better life, democracy, pluralism and stronger national unity".
In Damascus, Syrian Prime Minister Mohammed Naji Otri congratulated Mr Maliki on forming his government and reaffirmed "Syrian support for Iraq and its wish to accord all possible help".