The attack killed the Burundian deputy commander of the 5,000-strong Amisom and lightly wounded his boss, who was preparing to host negotiations in the compound when the blasts went off.
Seven of the deaths came as a result of a counterstrike from the AU base.
France evacuated 17 people wounded in the attack, the French foreign ministry said Friday.
A French military aircraft flew the 17 to Nairobi, where they were hospitalised, a ministry statement said.
Sheikh Ali Mohamud Rage, an spokesman for the anti-government al-Shabab group, claimed responsibility on Thursday for the attacks.
He said they were to avenge the death of Salah Ali Saleh Nabhan, a Kenyan-born al-Qaeda suspect, who was killed in southern Somalia on Monday during a raid by US special forces.
"We have got our revenge for our brother Nabhan. Two suicide car bombs targeting the AU base, praise Allah," Rage said.
"It took place at noon on the 27th of Ramadan, the best blessing. We knew the infidel government and AU troops planned to attack us after the holy month. This is a message to them."
Nabhan, 28, had been allied with al-Shabab, which the US accuses of being al-Qaeda's proxy in Somalia.
Witnesses said two vehicles with UN markings entered the coastal airport base of the Amisom mission - comprised of troops from Burundi and Uganda - unchallenged before blowing up.
Thegovernment of President Sheikh Sharif Ahmed said on Friday that al-Shabaab had six more stolen UN vehicles primed as suicide bombs.
Sheikh Yusuf Mohamed Siad, the state defence minister, said the fighters had seized more UN vehicles in recent months.
He said, however, that the bombings would not stop the government from launching fresh attacks against al-Shabaab.
The bombings came just hours after al-Shabab issued demands in return for the release of a French security consultant the group is holding hostage, including an immediate end to French support for Somalia's fragile government.
The French hostage is one of two security consultants kidnapped in Mogadishu in July.
Al-Shabab said the attacks were revenge for the killing of Nabhan [AFP]
His colleague managed to escape on August 26.
In return for his release, al-Shabab demanded the "immediate cessation of any political or military support to the apostate government of Somalia and the withdrawal of all its security advisers in Somalia", the group said in a statement.
They demanded the withdrawal of AU troops supporting Ahmed's administration and the departure of French warships trying to stamp out piracy in Somali waters.
The group's statement also called for the release of prisoners in other countries to be named later.
Fighting in Somalia has killed more than 18,000 civilians since the start of 2007 and left another 1.5 million homeless.