At least 14 people and nine fighters killed after simultaneous attacks on Afghan forces in southern military compound.
Taliban forces on Tuesday night attacked buildings housing government employees and military bases at the airport, which is used by Afghan, US and NATO military forces.
Attackers holed up in a building during the assault took family members of a military officer as hostages.
Afghan Special Forces deployed to the scene were moving slowly to retake the airport as any fast movement could cause civilian casualties during the hostage situation, foreign media reports.
Nearby businesses were torched.