Matthew Cunningham, 25, and Imogen May, 24, took responsibility for the direct action on the property in Victoria Street in Westminster.
The organisation said it spattered the department with paint after it issued more than 100 new oil and gas licences in the UK.
The targeting of the Department for Energy Security and Net Zero was one of a number of protests carried out by Just Stop Oil in London on Wednesday.
According to the Metropolitan Police, slow-marching demonstrators were cleared by officers from Westminster Bridge, Victoria Street and Vauxhall Bridge Road, as well as from roads in Marylebone and Kensington.
It is illegal criminal damage and I will leave that to the authorities
Mr Shapps, responding to questions about the paint protest on LBC during a round of broadcast interviews, said: “It is illegal criminal damage and I will leave that to the authorities.”
A video posted on Twitter by JSO appeared to show two protesters using a spray canister to spread orange paint over glass panels covering Mr Shapps’ Whitehall department.
Mr Cunningham, in another video posted by JSO, said: “This department oversees our energy policy and we know from last month that the Climate Change Committee, the Government’s independent watchdog for climate change policy, denounced the Government’s efforts against climate change.
“They said that they were far less sure that the Government would achieve net zero than it had been just one year ago.
“And they specifically said that the Government needs to stop issuing new oil and gas licences because it is not compatible with a net zero future.”
Ms May said the department is “failing on their only purpose” and that the climate situation is “getting really dangerous”.
Conservative Party chairman Greg Hands, posting a picture on social media of Westminster Bridge being blocked by JSO supporters, called the demonstration “unacceptable”.