Simon Fenton and his wife Sally, from Newbury, Berkshire, found the 17-year-old boy stowed in the roof box on Saturday after travelling home from Calais, France.
Mr Fenton said he saw two feet sticking out of the box and spotted the boy.
Thames Valley Police said a 17-year-old had been arrested on suspicion of immigration offences.
Describing the discovery, Mr Fenton said: “We had been home for about 10 minutes when my wife Sally told me to come quick.
“And there are two feet sticking out from the side of the roof box and thumping coming from inside, and neither of us registered what was going on.”
“And then a little hand came out and we saw somebody in there.
“That was a bit of shock to think they must have been in there since Calais.”
The couple, who had been holidaying in Dordogne, France, had stopped overnight on Friday at a hotel in Calais.
They had parked their Skoda on the street, as their car – with the roof box attached – was not able to fit in the hotel’s car park. The Fentons then caught a ferry from Calais to Dover, before returning home.
They then called the police after realising a boy was trapped in the roof box. They said police had warned them to stand back in case the boy had a handgun.
Mr Fenton said the boy told him he was from Egypt.
He also claimed his roof box had been emptied of their belongings.
He added: “He looked happy to get out and content police were there. I guess it was mission accomplished as far as he was concerned.
“I have a lot of sympathy for him actually and I don’t begrudge him for getting into our country, just it was a bit of a shock and have not had that much adrenaline pumping through our veins for a long time.”
Thames Valley Police said the investigation was now being dealt by immigration officials.
The Home Office confirmed an Egyptian male has been referred to them and is now in the care of social services due to his age.