The Conservative peer, who served in the Thatcher Cabinet from 1981 to 1987 in various roles, including as Tory Party chairman, said the instinct at the time had been to protect the political Establishment rather than delve into uncomfortable allegations.
His incendiary claim followed the admission by the Home Office that more than 100 files relating to allegations of historic child abuse over 20 years had gone missing.
It also followed reports that Lord Brittan, the former Tory Home Secretary, had been questioned by police under caution about an allegation he raped a woman in 1967, an allegation it is understood he has vehemently denied. It is also reported that an unnamed Labour peer is being investigated over claims of historic child abuse.
Lord Tebbit said: "At that time, most people would have thought that the Establishment, the system, was to be protected and if a few things had gone wrong here and there, it was more important to protect the system than to delve too far into it. That view was wrong then and it is spectacularly shown to be wrong because the abuses have grown."
Asked if he thought there had been a big political cover-up in the past, the former Secretary of State replied: "There may well have been," adding: "It was almost unconscious; it was the thing that people did at that time."
After a request from the Prime Minister, Mark Sedwill, the top civil servant at the Home Office, said he would appoint a senior legal figure to review what happened to a dossier on alleged paedophile activity at Westminster which in 1983 was passed to then Home Secretary Lord Brittan by the Tory MP Geoffrey Dickens.But critics insist public confidence can only be restored by a fully transparent and independent inquiry.
After Mr Sedwill revealed a Whitehall review last year had found 114 potentially relevant files from the period 1979 to 1999 were "presumed destroyed, missing or not found", Keith Vaz, the Labour chairman of the Commons Home Affairs Committee, said the Home Office appeared to have been losing files on an "industrial scale". Mr Sedwill is due to appear before the committee tomorrow.
Cabinet Office Minister Francis Maude stressed the first thing to do was to allow the review by the senior legal figure to be conducted quickly.
He said: "It may well be the answer then is to have a much broader inquiry, but it is too soon to come to that conclusion."
Earlier Deputy Prime Minister Nick Clegg said ongoing police inquiries had to take precedence but did not rule out a wider inquiry later. Michael Gove, the Education Secretary, was clear the Coalition would not support an independent inquiry.
But Labour backbencher Tom Watson launched a petition calling on Mr Cameron to "make amends for historic failures" by establishing a national inquiry into allegations of organised child sex abuse.
Yvette Cooper, the Shadow Home Secretary, upped the pressure, calling for a wide-ranging review in a letter to Theresa May.
She told the Home Secretary: "The scope of this investigation must look at how the Home Office, other parts of Whitehall, the police and prosecutions agencies handled allegations … but it also needs the flexibility to follow the evidence.
"Any stones left unturned will leave concerns of institutional malaise, or worse a cover-up, unaddressed."