The man, who was apparently alone, was "very quickly overpowered and arrested" and nobody was hurt, the statement added.
The incident happened at 11:30 pm (2130 GMT) Saturday night, about half an hour before the Tower normally stops admitting visitors.
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, AFP reported.
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 colors of PSG.
A screen near the bottom of the tower displayed a message welcoming Brazilian star Neymar to the club.
France has been under a state of emergency since the November 2015 attacks 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.