Initial reports said 34 people had died but a survey in nine hospitals indicated 42 were killed according to the French News Agency AFP.
Ahmad Ibrahim told a news conference 26 of the dead were civilians and eight were policemen.
Sixty-five policemen and 159 civilians were wounded in the blasts that rocked the city.
Bombers struck at least four times in Baghdadâ€™s morning rush hour, near the Red Cross headquarters and police stations in the capitalâ€™s bloodiest day since Saddam Husseinâ€™s overthrow.
At least two of the morning explosions appeared to have been suicide bombings, with an ambulance used in the Red Cross attack.
The explosions, sirens and smoke plunged Baghdad into fear and chaos at the outset of the Muslim fasting month of Ramadhan.
The onslaught â€œis not only criminal, itâ€™s sacrilegious,â€ U.S. Brigadier General Mark Hertling told reporters.
The International Committee of the Red Cross expressed outrage at Mondayâ€™s unprecedented suicide bombing of its Baghdad headquarters which killed at least 10 people and said it was weighing a withdrawal from Iraq.
It was the first time the Swiss-based relief agency, which for 140 years has sought to protect the victims of war, had been targeted by suicide bombers, although a number of officials have died in shootings and other attacks in places such as Chechnya and Afghanistan in recent years.
â€œWe are deeply shocked...because it is an attack against the ICRC...and that means, of course, a deliberate attack against our protective emblem and against our work,â€ chief spokeswoman Antonella Notari told Reuters Television.
The organisation issued a statement condemning the bombing, in which two of its Iraqi guards were among the dead, and stressed that all deliberate attacks on civilians were violations of â€œinternational humanitarian law and negated the most basic principles of humanity.â€ Most of the victims were passers-by.
Eye witnesses said an ambulance packed with explosives and bearing the ICRCâ€™s distinctive red cross emblem was used in Mondayâ€™s attack, which blew out the front of the building.
It was one of four rush hour bombings around Baghdad, including two strikes against police stations.
In all, at least 33 people died and many more were injured in the capitalâ€™s bloodiest day in months.
Notari said it was too early to say whether the ICRC would have to pull out of Iraq, where it has been present since 1980.
In that time, the country fought an eight-year war with Iran.
Bombs kill 42 in Baghdad