Iran-allied Houthi rebels and local militiamen traded tank and mortar salvos in the Khor Maksar district around the airport perimeter throughout the night and a Saudi-led coalition bombed Houthi positions from the air.
Residents said dozens of families fled, braving Houthi sniper fire and checkpoints as homes were shelled and burned.
"The scene is disastrous, not just in the streets where fighting is going on but inside houses where families are often trapped and terrified," local activist Ahmed al-Awgari said.
"Women and children have been burnt in their homes, civilians have been shot in the streets or blown up by tank fire," he added.
Scores of residents and fighters from both sides have been killed throughout the conflict, and residents said at least six Houthis and two local militiamen were killed overnight.
The Houthis hail from Yemen's far north and belong to the Zaydi sect of Shi'ite Islam.
They swept into the capital Sanaa in September and pushed south and east, saying they were winning a revolution against Sunni militants and corrupt officials.
But their push into Aden's outskirts on March 25 triggered a Saudi-led air campaign to drive them back and aid local gunmen.
The Sunni kingdom believes the group is a proxy for Shi'ite Iran, and Saudi backing for the resistance in Yemen's mostly Sunni south has raised fears that Yemen could descend into all-out sectarian war.
Yemen's vice president in exile, Khaled Bahah, said that the breakdown of Yemen's military during the conflict was helping al-Qaeda in the Arabian Peninsula (AQAP), the boldest branch of the global militant group, and said the Houthis were not confronting them.
"All we're seeing is them making war on the Yemeni people, not al Qaeda," Bahah told Qatar-based al-Jazeera TV.
"Al Qaeda has advanced and taken some positions in Yemen because of the security and military vacuum," he added.