A police official said the bomber rammed an explosives-laden vehicle into another car, setting off a huge blast that was heard across the coastal city.
A military spokesman for Somalia's Al-Qaeda-affiliated Shebab, Sheikh Abdul Aziz Abu Musab, confirmed that the militants had carried out the attack.
The sprawling airport area is a major base for members of Somalia's armed forces and houses several foreign embassies and African Union troops battling the Islamists.
It has been a frequent target of attacks by the Shebab, most recently in late December when the group launched a major assault against an African Union command centre.
Witnesses said they saw clouds of smoke after the explosion, and that security forces opened fire to disperse approaching onlookers -- prompting fears that another coordinated attack was under way.
Several witnesses also said they saw up to five destroyed vehicles in the vicinity of the explosion.
"There was a terrible explosion. The security forces have cordoned off the area. They opened fire to disperse people nearby," said Ali Suleyman, a witness.
The Somali government said the Shebab's intelligence chief, Abdishakur Tahlil, was killed in Monday night's raid.
The Shebab's former leader Ahmed Abdi Godane was also killed by a US air strike in September.
The organisation emerged from the Islamic Courts Union that controlled Mogadishu in 2006 before being pushed out by Ethiopian forces.
The militants were finally driven from their fixed positions in Mogadishu in 2011, and have lost several strongholds in the south and centre of the country in a recent offensive by the AU's 22,000-strong AMISOM force.
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