The climate change campaign group Just Stop Oil says it expects its two supporters to remain on the Queen Elizabeth II Bridge for 24 hours.
Police received reports of protesters scaling the bridge’s masts at the Dartford Crossing at 03:50 BST.
Diversions for traffic using the M25 east of London were put in place via the Dartford Tunnel.
Just Stop Oil said it was part of a protest against new government oil and gas licences.
The 1.7-mile (2.8km) bridge takes traffic southbound on the A282, which connects the M25 London Orbital Motorway from Essex to Kent. The Dartford Tunnel usually carries traffic northbound.
Essex Police’s Chief Constable, Ben-Julian Harrington, said: “It is a really dangerous and difficult environment.
“I think they’re crazy. I’m really frustrated.”
The force said the incident could take “some time to resolve due to the complexities of safely getting people down from height”.
“We expect the bridge to remain closed this evening and into tomorrow morning’s rush hour,” police said on Monday night, advising motorists to use an alternative route if possible.
Earlier, National Highways East said on Twitter there were “delays of two hours in both directions, with eight miles of congestion anti-clockwise and seven miles of congestion clockwise”.
Just Stop Oil said the two supporters ascended the two 84m (275ft) masts at the northern end of the bridge.
A group spokesperson said it was protesting against new government oil and gas licences and that “it is expected that the bridge will remain shut for at least 24 hours”.
The group said one of the pair, Morgan Trowland, was a 39-year-old bridge design engineer from London.
Mr Trowland said: “Our government has enacted suicidal laws to accelerate oil production – killing human life and destroying our environment.
“I can’t challenge this madness in my desk job, designing bridges, so I’m taking direct action.”
The Conservative MP for Dartford in Kent, Gareth Johnson, said it was “wrong” for the activists to “impose their views” on people trying to get to work and school.
“I totally deplore this action which is yet again causing traffic misery to people in and around Dartford,” he said.
“It is yet another example of how utterly selfish these protesters are.”
Pictures showed the two demonstrators had unfurled a pink Just Stop Oil banner.
Rob Hollyman, director of Youngs Transport Ltd in Purfleet on the north bank of the river, told BBC Essex: “It’s absolutely dreadful. It’s costing us hundreds of thousands of pounds.
“I’m sitting in a traffic jam instead of being at work running the business.”
The group said more than 450 of its supporters have been arrested during the last two weeks of action.
Conservative Home Secretary Suella Braverman has described the actions of demonstrators as “self-defeating” and “completely indefensible”.
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