The suspect who was arrested under the Terrorism Act had reached for a four-foot sword upon being challenged by unarmed police officers after deliberately driving at them outside the world-famous landmark.
Police said the man, who comes from Luton some 30 miles (50 km) north of London, had repeatedly shouted “Allahu akbar” (God is greatest) and was incapacitated with CS gas.
In an earlier statement, they said searches were ongoing in the Luton area and that they believed it was a lone-wolf attack.
“We believe the man was acting alone and we are not looking for other suspects at this stage,” it said.
Three officers were lightly injured while detaining the man, two of whom required hospital treatment.
Since March, 35 people have been killed in jihadist attacks in Britain, two of them in London and a third in Manchester.
Two of those involved a vehicle ploughing into pedestrians while the third was a bomb attack outside a pop concert by US star Ariana Grande in Manchester that killed 22 people, a third of them children.