A man in his 60s suffered "serious multiple lower limb injuries" and a boy had head and abdomen injuries following the incident, the ambulance service said.
Fire crews cut another man from the wreckage, pronounced dead at the scene, the BBC reported.
Police said the three people on board were from a family in Lancashire.
The aircraft reportedly crashed and flipped on its approach to the runway at the airfield in Dinas Dinlle.
The two casualties were taken to hospital at Ysbyty Gwynedd, Bangor. Their condition has been described as critical.
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An AAIB spokesman said: "We are aware of the incident and have dispatched a team to carry out a preliminary inspection."
North Wales Police said family liaison officers are currently supporting the family.
Chief Inspector Darren Wareing said: "An investigation is on-going and we are appealing to anyone who witnessed the plane as it approached the runway of Caernarfon Airport to contact police.