Loud blasts and gunfire were heard at the airport shortly after the plane arrived, witnesses said.
The cabinet members including Prime Minister Maeen Abdulmalik, as well as Saudi Ambassador to Yemen Mohammed Said al-Jaber, were transferred safely to the city's presidential palace, the witnesses and Saudi media said.
"We and the members of the government are in the temporary capital of Aden and everyone is fine," Maeen tweeted from Maasheq palace. "The cowardly terrorist act that targeted Aden airport is part of the war that is being waged against the Yemeni state and its great people."
A local security source said three mortar shells had landed on the airport's hall.
The newly formed cabinet unites the government of President Abd-Rabbu Mansour Hadi with southern separatists. The two groups are the main Yemeni factions in a southern-based, Saudi-backed alliance, fighting against the Iran-aligned Houthi movement that controls the north, including the capital Sanaa.
Live TV footage from Saudi-owned Al Arabiya channel showed dozens of people leaving the airplane when a first blast hit the airport's hall. Heavy gunfire from armoured vehicles followed with plumes of white and black smoke rising from the scene.
Other video showed damage to the terminal's concrete walls and smashed glass.
The southern port city of Aden has been mired in violence because of a rift between the separatists and Hadi's government, based there after being driven from the capital by the Houthis in 2014.
The separatist Southern Transitional Council (STC), which seeks independence for south Yemen, declared self rule in Aden earlier this year, triggering clashes and complicating U.N. efforts to forge a permanent ceasefire in the overall conflict.
The Saudi-led coalition announced earlier this month the new power-sharing cabinet that would include the separatists.The cabinet landed from Riyadh where both parties negotiated for more than a year with Saudi mediation.