The two individuals, aged 20 and 32, turned themselves in on Sunday night, according to a statement issued by the Leipzig police.
The younger man said through a lawyer that he lost control of the car while driving in the airport area the previous night and crashed through the fence.
He then left the scene of the accident and sent for the car to be towed away.
He did not have a driver's licence.
The 32-year old admitted he was the owner of the car and gave no further details, the police said.
Security sources told the DPA news agency that the two men had been practising for a driving test near the airport on Saturday evening when they lost control over their vehicle, damaging the fence.
The car has been seized and the investigation was continuing, the statement also said.
German authorities had been investigating how a gaping hole was cut into a fence around Leipzig airport.
The damage was discovered about 11pm on Saturday.
Police said initial evidence suggested that a vehicle had ploughed through the fence along a publicly accessible road.
Emergency teams that searched the airport grounds did not find the vehicle or any intruder.
In August, an attack with an explosives-laden drone took place at the airport.
German authorities blamed that on Russian authorities, an allegation denied by the Kremlin.
with DPA
