The pilot survived and was taken to Kathmandu Medical College Hospital for treatment, said a doctor at the hospital who was not authorised to speak to media.
The pilot had eye injuries but was not in any danger, the doctor said.
The Saurya Airlines plane on Wednesday was heading from Kathmandu to the resort town of Pokhara.
It was not clear how it slipped.
Local media images showed smoke rising and plane wreckage scattered all over a ditch.
A fire was brought under control.
Tribhuvan International Airport, the main airport in Nepal for international and domestic flights, was closed as emergency crew worked.
It is monsoon season in Kathmandu but was not raining at the time of the crash.
Visibility was low across the capital, however.
Saurya Airlines operates the Bombardier CRJ 200 on domestic routes.
In 2019, a Bangladeshi airliner crashed at Tribhuvan airport, killing 51 people while 20 on board survived.
An investigation confirmed the plane was misaligned with the runway and its pilot was disoriented and tried to land in "sheer desperation" when the plane crashed.
In 2015, a Turkish Airlines jet landing in dense fog skidded off a slippery runway at the airport.
The plane was carrying 238 people but there were no serious injuries.
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