Top Whitehall civil servants are even posher today than in the 1960s, says the Social Mobility Commission (SMC).
Hidden rules favour those with the "right accent", a report - Navigating the Labyrinth - claims.
Almost three-quarters (72%) of those in senior posts are from privileged homes, up from two-thirds (67%) in 1967.
The survey of more than 300,000 civil servants found a "behavioural code" that makes it harder for people from working-class homes to get promoted.
Only 18% of senior civil servants are from a low socio-economic background, says the SMC, the body appointed by government to advise on social mobility,
Most of the 445,000 civil servants are employed by government departments.
Jobs range from administrative and operational roles such as prison officers and tax collectors, up to permanent secretaries, the most senior grade, who work with ministers to run their departments.
Among those top Whitehall mandarins, 59% attended a private school - three times the rate among the wider population.
The permanent secretary at the Department for Transport who leads on social mobility, Bernadette Kelly, has welcomed the report.
"It chimes with much of my experience in talking to colleagues right across the civil service, and at all grades, about social mobility," she says.
"The sense that there is a hidden route to the top that those from more privileged backgrounds find easier to navigate."
Those from advantaged backgrounds are defined as having parents in professional or managerial occupations.
The parents of those defined as coming from disadvantaged backgrounds had either never worked, or had jobs such as drivers, cleaners, receptionists or mechanics.
There is a culture that favours polish over performance, creating a "class ceiling" that prevents people from poorer backgrounds getting the top jobs, the report finds.
One civil servant told researchers how senior staff would sometimes break into Latin.
"I know that is a bit of a stereotype, but it is so real," Kristine says.
"You'll be in a ministerial meeting and they'll sort of talk in Latin, but they're making what you'll realise later is a sort of joke about Brussels that everyone sort of understands, and laughs."
The report identifies how "cosy fireside chats" conducted in a neutral received pronunciation accent help those with the right background navigate the "velvet drainpipe" of recruitment.
"There is a definite style of speaking," one interviewee tells researchers in the report.
Another civil servant says people with working-class accents tend to fare less well: "I used to put on a bit of an accent or try to enunciate a little bit better," Pauline says. "Ridiculous and humiliating, when I look back on it."
The report finds those from advantaged backgrounds are much more likely to work in policy jobs in Whitehall, benefiting from "the propulsive power of being in London".
Two-thirds of civil servants in the capital come from privileged backgrounds, compared with 41% in north-east England.
The government has plans to reform the civil service, initiated by the prime minister's former adviser Dominic Cummings, but this has been delayed by the pandemic.
Mr Cummings said he wanted to encourage "weirdos and misfits with odd skills" to work in government.
One ambition is to move more officials out of London, but the Social Mobility Commission warns the proposal to relocate 22,000 civil servants away from the capital risks increasing the concentration of civil servants from advantaged backgrounds still further.
"The civil service should be a role model for fair recruitment and career progression," says the report.
The authors accept that the civil service is acutely aware it needs to do better, and has made progress on diversity and inclusion.
However, getting in is not enough and more action is needed to nurture career progression, the report says.
"When I get feedback, one of the things [staff from disadvantaged backgrounds] raise is the degree to which the conversation is all about politics and about people on Twitter that everyone's following," civil servant Alistair tells the researchers.
"You know, the majority of the country are not reading these tweets. Probably the entire audience for this tweet that we're discussing at the moment is in this room!"