The offensive, dubbed Operation Golden Arrow, is the first major advance by the loyalists since the Huthi rebels entered the port city in March, forcing President Abedrabbo Mansour Hadi into exile in neighbouring Saudi Arabia.
Despite an appeal from US President Barack Obama to King Salman of Saudi Arabia for an urgent end to the fighting, Saudi-led warplanes carried out six raids on rebel positions before dawn, witnesses and military sources said.
Popular Resistance fighters -- a southern militia that has been the mainstay of support for Hadi -- recaptured the provincial government headquarters in the Mualla district by Aden's main commercial port, militia spokesman Ali al-Ahmadi told AFP.
Yesterday, the militia, backed by reinforcements freshly trained and equipped in Saudi Arabia, retook the airport and much of the surrounding Khormaksar diplomatic district.
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"After the recapture of Khormaksar, there was a collapse in the ranks of the Huthis and their allies," renegade troops loyal to Hadi's predecessor Ali Abdullah Saleh, Ahmadi said.
It was the defection of the 39th Armoured Brigade on March 25 that had enabled the rebels to take the airport.
Much of Aden has been reduced to rubble by four months of ferocious fighting.
At least 12 civilians were killed and 105 wounded, Aden health department chief Al-Khader Laswar told AFP.
Eight loyalist militiamen were killed and 30 wounded in the fighting, Laswar added.
There was no immediate word on rebel losses.
The rebel offensive comes after the failure of a UN-declared truce that was to have taken effect just before midnight on Friday.
UN chief Ban Ki-moon had announced a six-day ceasefire to allow the delivery of desperately needed relief supplies.
Ban was "very much disappointed" by the failure of the truce, his spokesman said.
The two leaders "spoke about the urgency of stopping the fighting in Yemen and the importance of ensuring that assistance can reach Yemenis on all sides of the conflict".