The supporters of the separatist Southern Transitional Council pulled out of the headquarters of Hadi's government, the supreme court and the central bank as well as Aden's main hospital, Muammar al-Iryani said on Twitter.
Hadi's government has refused to enter talks with the separatists until they withdraw from the positions they seized in deadly fighting last week.
Iryani said preparations were also underway for a pullout by the United Arab Emirates-trained fighters from the interior ministry and the Aden oil refinery.
The start of the withdrawal had been announced by the Saudi-led military coalition which has been backing Hadi's government against northern-based Houthi rebels since 2015.
The UAE is Saudi Arabia's main partner in that coalition but trained and equipped the separatists as part of its efforts to consolidate control of the south.
The fighters of the so-called Security Belt last week ousted unionist forces loyal to Hadi from what was the capital of the formerly independent south in clashes that left around 40 people dead.
The coalition condemned the takeover of the city and urged the Security Belt to withdraw to clear the way for peace talks.
On Thursday, a joint Saudi-UAE military delegation travelled to Aden to discuss a the details of a separatist pullback.
AFP journalists saw Saudi and Emirati military vehicles deployed around the positions vacated by the separatists on Saturday.
The presidential palace was also placed under Saudi and Emirati protection, they said.
The coalition said the positions that had been occupied in Aden would be handed over to the government.
Several other installations, including military camps, remain under the control of the separatists, however, and it was not immediately clear if they too would be vacated as part of the withdrawal.
The Yemeni embassy in Washington tweeted a foreign ministry statement welcoming the Saudi initiative to address the "coup" in Aden.
It said the separatists "must first commit to total withdrawal from areas forcibly seized by STC in past few days before start of any talks".
South Yemen was an independent country until it merged with the north in 1990. An armed secession bid four years later ended in occupation by northern forces, giving rise to resentments which persist to this day.
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