The attacks, which wounded dozens more, came as security forces battle militants in the western province of Anbar, including the Islamic State of Iraq and the Levant (ISIL), a powerful jihadist group that has exploited the chaos in neighbouring Syria.
With violence at its highest level since 2008, diplomats have urged the Shiite-led government to reach out to Sunnis in order to undercut support for militancy, but Prime Minister Nuri al-Maliki has taken a hard line ahead of April's parliamentary elections.
The three explosions killed 25 people and wounded another 35, security officials and a medical source said.
One attack was just opposite the foreign ministry, but accounts differed as to what caused the explosion. Two security officials said the blast was caused by a car bomb, but witnesses said a suicide bomber was responsible.
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The area surrounding the ministry has been hit by explosions in the past, notably in August 2009, when a massive truck bomb devastated the building, and again ahead of an Arab summit in Baghdad in 2012.
Blood and pieces of flesh littered the scene at the restaurant, and soldiers said one of their comrades had wrapped his arms around the bomber in a bid to save others.
Later in the afternoon, three car bombs in southeast Baghdad killed eight people and left 32 more wounded, while a rocket attack on Haifa street in the centre of the capital wounded five.