Lord Triesman, who was the initial chairman of England's bid to host the 2018 World Cup, has previously claimed four Fifa members sought bribes in return for votes.
In the light of recent media allegations of corruption in Qatar's successful bid for the 2022 World Cup, Lord Triesman stepped up his attack on Fifa, claiming attempts by the organisation's president Sepp Blatter to dismiss the issue was a tactic that would have been approved by movie Godfather Don Corleone.
"Fifa, I'm afraid, behaves like a mafia family. It has a decades-long tradition of bribes, bungs and corruption," Lord Triesman said.
"About half of its executive committee who voted on the last World Cup have had to go.
"Even its past president Joao Havelange has been removed from his honorary life presidency in his 90s.
"Systematic corruption underpinned by non-existent investigations where most of the accused are exempt from the investigation make it impossible to proceed.
"Foreign construction workers dying in their dozens in Qatar stadium construction sites are essentially ignored."
Speaking during debate on the Queen's Speech in the House of Lords, he applauded the stand taken by current FA chairman Greg Dyke against the "grotesque" accusation by Mr Blatter that criticism was racist.
He told peers: "Don Corleone, I believe, would have recognised the tactics and he probably would have admired them."
Lord Triesman also hit out at the governance of football in England by the FA.
"The fit and proper persons test failed to stop a Thai politician, a former prime minister with a notorious human rights record, from acquiring a major club in this country (Manchester City) or unknown owners controlling other great clubs that are essential to the sporting culture.
"It is important to our sports culture and it is important to fans."
He added: "Governments should stay out of sport, but they are entitled to designate the lead associations to regulate each sport and to set out the regulatory requirements and standards in the conduct of international sport.
"In the case of the FA, it will be a daunting task to take even the most modest steps, but it is time for the organisation and other sporting organisations to step up to the mark.
"Let's try to eliminate corruption wherever it exists, not least in those things that are so dear to people's hearts."