Air traffic controllers saw the single-engined plane plunge into the runway while coming into land at Kidlington air training centre, north of Oxford.
The pilot, who was in his thirties, was killed instantly, police said.
His passenger, thought to be an instructor in his forties, was taken to the John Radcliffe Hospital in Oxford with multiple injuries. He died shortly afterwards in the intensive care unit.
The crash - at 5.30pm - happened less than 100 yards from airport buildings, moments after the pilot reported an engine fault to air traffic control.
An investigation was launched by the Civil Aviation Authority.
An Oxfordshire fire service spokesman said: 'The plane developed an engine fault of some sort as it came into land following a training flight and crashed on to the runway.'
A spokesman for Thames Valley police said the identities of the two men were not being released at this stage.
Ken Meehan, principal of the airport's flying school, CSE Aviation Ltd, said: 'It was very busy with jets and helicopters coming and going to Silverstone.
'Basically, if people can afford it and want to avoid the traffic jams, they fly into the airport as a means of getting to the race track.'
He said the four-seater Piper Cadet was one of a fleet of 32 aircraft at the flying school.
Jim De Salis, the operations manager, refused to comment on the cause of the crash while the CAA investigation was in progress.