He was photographed by Laura Martin, 22, who was a passenger in her father's van when they pulled up next to the learner vehicle near Norwich City's Carrow Road stadium.
The self-employed instructor was in a 5Day car when he was seen snoozing.
The firm said in a statement his behaviour was "unacceptable" and he had been "immediately dismissed".
It said it contracted work to self-employed instructors who provided their own vehicles and were also able to carry out their own private lessons.
"We cannot be 100% certain this particular instructor was working for us at the time the picture was taken. However, we expect high standards of tuition from all our contractors," said 5Day.
"We found this behaviour unacceptable and the individual in question was immediately dismissed and will no longer receive any work from our company.
"We have many instructors working for us weekly around the UK and this is the first instance of this type of behaviour."
Miss Martin said: "The instructor was asleep for the entire time [we were next to him]. The only thing that woke him up was my dad revving the van very loudly to see if that would wake him up.
"As for the pupil he was teaching, he looked extremely flustered."
She posted the photo on her Facebook page and it has attracted a string of comments.
"Not so funny if the poor boy had caused an accident and killed himself or someone else," said one person.
Norfolk Police confirmed it had received a complaint from a member of the public about the unnamed instructor.
Ch Insp Chris Spinks, head of road policing in Norfolk and Suffolk, said: "There is no specific offence that covers an individual supervising a learner driver being asleep as there is with them being drunk or using a mobile phone.
"However, a supervisor or instructor can be seen as being in charge of that vehicle so should be in a position to intervene if required.
"Failure to adhere to this responsibility could therefore make the supervisor liable to an offence in the event of a collision or other incident."
A driving instructor who was photographed asleep in a car with a learner at the wheel has been sacked.