"The man was arrested on suspicion of being involved in the commission, preparation or instigation of terrorism at 10:15hrs (0915 GMT) today and he has been taken into custody," the statement said.
The 30-year-old man was arrested at an address in West London and police are searching the property as part of the inquiry.
A warrant for the extended detention of the 26-year-old man involved in Friday's incident has also been granted, the statement said.
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.
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