Larry has been in residence at Downing Street since 2011, and Palmerston was brought in to keep rodents at bay in the Foreign Office earlier this year.
Gladstone, who is thought to be 18 months old, is a former stray adopted from Battersea Cats and Dogs Home.
Larry was one of the few post holders to survive Theresa May's government reshuffle in July.
It was confirmed he would stay on after the departure of David Cameron, who took the opportunity in his final prime minister's questions to quash rumours he did not get along with the mouser.
Relations between Larry and Palmerston are rumoured to have been strained, and there was speculation that Larry's recent trip to the vet was the result of one of their run-ins.
But the latest feline appointment - who is named after former Liberal prime minister and four-time chancellor William Ewart Gladstone - signalled a willingness to stand up to No. 10.
A caption on Gladstone's photo - taken of him in a cat carrier - reads: "The humans had to keep me in this cage in case I ran down the street and tormented some other mouser called 'Larry'.  Personally, I've never heard of him."
Asked why Gladstone, who was previously called Timmy, had been drafted in a spokeswoman said it was to "help control the mice problem in the 1 Horse Guard Road building".
A third cat has joined the Whitehall mouse patrol with the arrival of Gladstone at the Treasury.