Al Jazeera’s Rawya Rageh reports from Baghdad that the ten explosions took place outside as well as in the yards of Christian homes across Baghdad.
She said the attacks were "not simultaneous but clearly appeared to be coordinated".
The attackers used a combination of grenades and simple homemade bombs. In at least two cases, police arriving on the scene found additional unexploded bombs.
The attacks come just a few days after al-Qaeda renewed its threat against Iraq’s small Christian community during this holiday period.
The worst attack was in the central Baghdad district of Al-Ghadir, where a homemade bomb exploded, killing two people and wounding three others, an interior ministry official told the AFP news agency.
The Al-Ghadir area has a significant Christian population.
Two bombs exploded in west Baghdad, one in the garden of a home in Yarmuk, where one person was hurt, and another in Khadra, wounding two Christians.
Another was hurt in an explosion in the Karrada district, near a cathedral that had previously been attacked.
In south Baghdad, three Christians were hurt in explosions in Dora and two in Saidiya.
The casualties were confirmed by hospital officials. All of the officials spoke on condition of anonymity because they were not allowed to talk to reporters.
Major General Qassim al-Moussawi, a spokesman for the Baghdad operations command, put the toll at one dead, denying the victim was a Christian, and six wounded. He said two other bombs were defused before they exploded.
Thousands flee
"Christmas Eve and day had gone by without incident, but unfortunately that tense calm is now shattered," Rageh said.
Thousands of Christians are reported to have already fled the capital to the relatively safer north after a series of attacks targeting the community since October.
The most serious of those attacks was a deadly siege on a Syrian Catholic church in Baghdad, in which more the 50 worshippers were killed.
The government of Iraqi Kurdistan in the north of the country has invited Christians to take refuge there, while the local federal government in Baghdad has also said it will do all that it can to protect them, Rageh reported.