The violence broke out on Tuesday, after a day of peaceful protests, when riot police clashed with opposition supporters who claim that the elections held in May last year were rigged.
The shootings took place at Old Airport, a wealthy neighbourhood where many foreign expatriates live, according to doctors. About 700 Britons live in Ethiopia, including UN workers.
Sporadic gunfire was heard near the French and Dutch embassies, and protesters were seen throwing stones despite a heavy troop presence.
Those wounded included a 7-year-old girl who lost an eye after police hit her with a baton. An 11-year-old boy, Yarad Wubetu, was shot in the stomach when he came out of his home to watch police chasing a group of young men, according to his mother.
Mengistu Dagagnew, 24, said that he had been "shot while trying to enter my house". Lying on a hospital stretcher, he said: "The shooting was indiscriminate. They were attacking children and women."
The continuing crisis is a setback for Prime Minister Meles Zenawi, who has been hailed by Tony Blair as a member of a new generation of African leaders committed to democracy. He was a member of Mr Blair's Commission for Africa, along with the president of Tanzania, Benjamin Mkapa, whose country has also been beset by election-related violence this week.
However, the government of Mr Meles, a former hardline Marxist rebel, has shown little tolerance for dissent and has been accused of human rights abuses.
Dozens of activists, including opposition leaders, have been rounded up by police. A total 2,000 people have been arrested, according to diplomats. ActionAid International has reported its policy manager in the country, Daniel Bakele, is being held "in unacceptable conditions" after being arrested and detained without charge earlier this week.
Lord Triesman, the Africa minister, called the Ethiopian chargé d'affaires on Tuesday to express "concern and alarm" at the deteriorating situation. He appealed for restraint and for an urgent, independent investigation into the incidents to be carried out.
The opposition has accused the ruling party of rigging the 15 May elections that gave Mr Meles's Ethiopian People's Revolutionary Democratic Front control of nearly two-thirds of parliament. The opposition Coalition for Democracy and Unity has spearheaded the protests against the alleged poll-rigging since that time, accusing President Meles of clinging to power by fraud.
In June, after police killed 22 and arrested 600 anti-government demonstrators in an earlier round of protests, Mr Meles declared a state of emergency in Addis Ababa.
In a separate development, Ethiopia has been engaged in a new round of sabre-rattling on its border with Eritrea, triggering fears of a renewed border conflict.
Although Eritrea is being blamed for the latest rise in tensions, both sides have moved troops and military hardware closer to the border in the past 10 days.
"This potentially volatile situation could lead to a renewed outbreak of war," said Major-General Rajender Singh, commander of United Nations peacekeepers in Ethiopia and Eritrea.
"If not addressed, if we do not take measures, if the international community does not act, maybe the situation may deteriorate to a level where the worst can happen - and that is war."
The neighbours fought a bloody two-and-a-half year border conflict until 2000. The war was famously described by a senior UN official as "two bald men fighting over a comb".