The claim followed a setback the previous night when a warplane raided Islamist positions, killing 13 fighters, a Fajr Libya spokesman said.
If independent sources confirm the airport has changed hands, it would be a major defeat for the nationalist fighters from Zintan west of Tripoli who have held the airport since the fall of long-time dictator Moamer Kadhafi in 2011.
A statement shown on screen on An-Nabaa television, regarded as close to the Islamists, said: "Fajr Libya announces that it totally controls Tripoli international airport."
Later a spokesman for the Islamist coalition, partly comprising men from Misrata, east of Tripoli, said its fighters "have entered the airport and are mopping up pockets of resistance".
On the political front, Libya's outgoing provisional General National Congress (GNC), which was dominated by Islamists, will resume operations despite being superseded by an elected national parliament, its spokesman said on Saturday.
The strategic airport 30 kilometres (20 miles) south of the Libyan capital, has been shut since July 13 amid clashes between the Islamists and the Zintan force, allies of rogue general Khalifa Haftar, based at Benghazi in eastern Libya and hostile to the Islamists.
The Islamist coalition, which repeatedly claims successes against the nationalists, on Thursday organised a visit by Libyan journalists to an army base on the way to the airport, to prove they had taken it.
Fajr Libya on Saturday accused the United Arab Emirates and Egypt of involvement in the Friday night air raid and an earlier strike when two unidentified aircraft bombarded Islamist positions on Monday night.
"The Emirates and Egypt are involved in this cowardly aggression," the coalition said in a statement read out to Libyan journalists in Tripoli.
In the wake of the raids, the GNC will convene again despite being a national parliament being elected in June, its spokesman said.
Friday night's air strike killed 13 Islamists and left 20 wounded, Ahmed Hadia, a Fajr Libya spokesman, said, updating an earlier toll.
"We reserve the right to respond at the opportune moment," Hadia said.
The Islamist fighters believe Libya's provisional government and newly elected parliament "are accomplices to these raids and in doing so have committed an act of treason that removes their legitimacy to govern the people", he said.
The spokesman called on the GNC, whose mandate expired when the new parliament was sworn in earlier this month, to meet again to "defend the sovereignty of the Libyan state".
The Islamists were well represented in the GNC but non-Islamist blocs dominate the new parliament, which is holed up along with the provisional government in Tobruk 1,600 kilometres (1,000 miles) east of Tripoli, to avoid the violence in the capital.
GNC spokesman Omar Ahmidan said on a local television station: "The General National Congress will hold an emergency meeting in Tripoli to save the country's sovereignty."
The latest deadly air strike targeted an army base to the south of Tripoli and a nearby warehouse, Fajr Libya said earlier.
The drawn-out battle for the airport has sparked the worst violence in the Libyan capital since the 2011 uprising.
Fajr Libya's allegation that the UAE and Egypt were behind the raids followed days of wild speculation about the provenance of the warplanes that swooped over the airport on Monday.
Haftar claimed to be behind the first raid, but specialists doubted his ability to carry out such an attack.
The Islamists suspected foreign aircraft acting at the behest of the Libyan government after the new parliament elected on June 25 called for foreign intervention to protect civilians.
In neighbouring Egypt, new President Abdel Fattah al-Sisi is hostile towards Islamists, though authorities in Cairo have refused to comment.
On their way to the airport, Fajr Libya fighters seized from Zintan militiamen a Tiger armoured car developed by the UAE, television pictures showed.
The Islamists claimed the vehicle was proof of UAE support for the Zintan fighters.