After the arrest, police at the site checked the venue and called for it to be evacuated at around 00:30, 15 minutes before the Eiffel Tower normally closes, the statement added.
The would-be intruder was wearing a Paris Saint-Germain football shirt and at the time of the incident the tower was lit up with the team colours of PSG.
A screen near the bottom of the tower displayed a message to the club.
France has been under a state of emergency since the on Paris restaurants, a concert venue and a sporty stadium that left 130 people dead.
Since the first wave of terror attacks in January 2015, 239 people have been killed.