ISIS claimed responsibility for the attack in a statement posted by its Amaq News Agency.
At least seven people were killed and 12 were injured in the attack, according to a local security official.
The bomber rammed an explosives-laden vehicle into the entrance gate to the building, three Yemeni security officials said.
Emergency trucks rushed to the scene as smoke billowed in the sky just minutes after the attack.
"Fierce clashes between security forces and suspected terrorists escalated after the explosion," a local Aden security official said.
The fighting has since subsided, officials said.
Regional powers fight proxy war in Yemen
Yemen, a troubled nation that borders Saudi Arabia and Oman, is embroiled in a complex civil conflict
. Saudi Arabia and the United Arab Emirates support the central government in its fight against Iran-backed Houthi rebels, who toppled the internationally recognized leadership there in 2015. Saudi Arabia and its allies joined the war, leading a military operation to support the government, now based in Aden.
Saudi Arabia accuses Iran of arming the Houthi rebels, and has been fighting a proxy war in Yemen against Iran. The UN Human Rights Office says thousands have been killed
in the escalating violence in the past two years.
Meanwhile terrorist groups have flourished in the country. Yemen is home to al Qaeda in the Arabian Peninsula
, considered by US officials to be the most dangerous branch of the terrorist network.
Stephen O'Brien, the UN undersecretary-general for humanitarian affairs and emergency relief coordinator, said the country's citizens face a "triple threat" of "armed conflict, famine, and deadly disease that has already killed, injured, displaced or otherwise affected millions and it will spare no one if it continues unchecked."
Yemen's conflict is often called "the silent war"
because it receives relatively little attention.