All flights have been suspended after the incident happened at 3.40 local time on Saturday.
Police stopped him after several hundred metres, it said.
The news was announced by Hannover Police on Twitter, saying "An incident occurred at 15:40, after a man had broken through a gate with a auto and drove to the apron".
The man was quickly arrested and taken into custody. No one was hurt. The auto was stopped by federal police and the driver was detained.
Bomb disposal experts examined the vehicle but found no risky objects.
The man was believed to have acted alone.
Federal police said the driver, who is in his mid-20s, tested positive for amphetamines and cocaine.
The lockdown will reportedly last till at least 9pm local time, meaning that the ninth busiest airport in Germany will be out of operation for over five hours. Police said they couldn't give further information on where he came from because he wasn't carrying an identity card.
It wasn't immediately clear how long flight operations would be suspended.
The global airport in Hannover handled 5.87 million passengers a year ago, though it isn't one of Germany's top hubs.
A terminal at the airport was closed to passengers but the other two remained open. Authorities eventually towed away the vehicle and searched outside the airport for other possible threats.