Gun battles killed eight government troops, including a colonel, and four of the unidentified militants, medical sources told Reuters. No one immediately claimed responsibility for the attacks.
The government has struggled to keep order since its forces, backed by a Saudi-led alliance, took back Aden from largely Shi'ite Muslim Houthi fighters in July.
Tensions mounted further over the weekend after the Houthis joined Shi'ite power Iran in condemning the decision by Sunni Muslim power Saudi Arabia to execute a prominent Shi'ite cleric.
The city and surrounding province were now stable and the curfew was ordered as a precaution, said local authority spokesman Nizar Anwar.
Saudi Arabia has led an Arab coalition in a military campaign since late March to stop the Houthis, allies of Riyadh's main regional foe Iran, from taking complete control of Yemen after they seized the capital Sanaa in 2014 and advanced south.
The coalition announced on Saturday the end of a ceasefire that began on December 15 and has been repeatedly violated by all sides.
The Houthis say they are leading an Islamic-inspired revolution against corruption, but the Saudis have accused the group of acting as a proxy for Tehran.
ISIS and Al Qaeda branches in Yemen have also taken advantage of the chaos to step up operations.