Yemeni pro-government forces, backed by the United Arab Emirates, are reported to have stormed the airport serving the rebel-held Red Sea city of Hudaydah.
The UAE's state news agency, Wam, said troops had gained control of large parts of the facility and that dozens of Houthi rebels were dead or wounded.
They launched a major offensive last week to capture the city and its port.
The port is crucial to the delivery of aid supplies to Yemen, where 8.4 million people are at risk of famine.
There are grave fears that the fighting will disrupt the flow of this humanitarian assistance.
Yemen has been devastated by a conflict that escalated in early 2015, when the Houthis seized control of much of the west of the country, including the capital Sanaa, and forced President Abdrabbuh Mansour Hadi to flee abroad.
Alarmed by the rise of a group they saw as an Iranian proxy, the UAE, Saudi Arabia and seven other Arab states intervened in an attempt to restore the government.
Almost 10,000 people - two-thirds of them civilians - have been killed and 55,000 others injured in the fighting, according to the United Nations.
The coalition says the Houthis have used Hudaydah's port to receive weapons smuggled by Iran - an allegation both Tehran and rebel leaders have denied.
Wam reported on Tuesday that Yemeni and Emirati troops had entered the airport, in the south of the city, after they "destroyed the defences of the Iranian-backed Houthi militias".
Troops captured dozens of rebels and also found many dead and wounded, including field commanders, the news agency said. Yemeni military sources confirmed that the airport had been stormed.
The battle cause panic among residents. "My children are terrified. The fighting and the sounds of explosions are everywhere and we are stuck in our house... with no running water," Iman, a mother of two, told Reuters news agency.
First objective completed
By Paul Adams, BBC News
Hudaydah airport is the first strategic objective of pro-government forces and their Saudi-led coalition backers. Now the fighting may move north to the main Hudaydah-Sanaa highway, as the coalition seeks to cut off a vital Houthi supply line to the capital.
At the same time, aid workers based in Hudaydah itself say the fighting appears to be getting closer. They say people are trapped in their homes and running out of food. Some people displaced by clashes in outlying areas are seeking shelter in school buildings.
The Saudi-led coalition directing this assault says it's doing everything in its power to avoid making the humanitarian situation worse. As fighting moves closer to built-up areas, this will be a major challenge.
On Monday, UN spokesman Stéphane Dujarric said about 26,000 residents had fled their homes since the Hudaydah offensive began last Wednesday and warned that the number was expected to increase.
An aid worker from the Norwegian Refugee Council said he and his colleagues were increasingly worried for their safety as the fighting reached Hudaydah's southern outskirts.
"Even before this current crisis, people had difficulties finding food. People don't have anything left inside their houses here," Waleed Al Ghadhi said. "Now the situation has become dire."
The UN has warned that in a worst-case scenario, the battle for Hudaydah could cost up to 250,000 lives, as well as cut off aid supplies to millions elsewhere.
The city is one of Yemen's most densely populated areas, with an estimated population of 600,000. Its port is the principal lifeline for about two-thirds of Yemen's population, which is almost totally reliant on imports.
The UAE's Minister of State for Foreign Affairs, Anwar Gargash, said on Monday that the coalition would seek to minimise the risk to civilians by focusing on capturing the port rather than engaging in an "ugly fight" for the city.
He also expressed hope that UN special envoy Martin Griffiths would be able to persuade the Houthis to withdraw from Hudaydah and engage in peace talks.
Are you in Yemen? Share your experiences by emailing email@example.com.
Please include a contact number if you are willing to speak to a BBC journalist. You can also contact us in the following ways: