England assured themselves of participation in the finals next summer by defeating Poland 2-0 at Wembley on Tuesday night.
And Perryman says fans are convinced they will be safe when they make the trip over to South America.
There were anti-government protests on the Brazilian streets during the Confederations Cup in June, which have continued, and the Foreign Office website says of the country: "Levels of crime and violence are high, particularly in major cities."
But Perryman said: "There were riots on the streets of London 12 months before the London Olympics and there was no public disorder during the Olympics.
"Let's get this into perspective. Since 2006 the newspapers have told us we're going to be car-jacked and murdered whatever country we're playing in, whether it's South Africa or Brazil or Ukraine.
"We had a wonderful time in South Africa and Ukraine, and I'm sure we'll have a wonderful time in Brazil. They've got some social problems, get over it.
"The World Cup is the most prestigious event in the world to host after the Olympics and Rio hosts the Olympics two years after the World Cup. Why on earth would they let things go wrong?
"They will do their absolute utmost to ensure fans have a safe, enjoyable and wonderful World Cup. We had exactly the same stories in the build-up to World Cup 2010.
"People aren't stupid. There are parts of London I wouldn't suggest a tourist walks down at the dead of night, and there are parts of Rio where I'm sure it's not a good idea to walk on your own in the dead of night. But that's part of travelling to different countries."
Meanwhile, travel expert Simon Calder has admitted England fans will have to dig deep to make the journey to football's premier event.
Calder said: "Within an hour of the final whistle at Wembley last night one of the big sports tour companies had come out with a £7000 deal which actually, I think is a pretty good price."