Awake and likely startled, the pilots manoeuvred the aircraft around and landed on the runway 25 minutes later.
Incredibly, no one was harmed and the plane landed safely.
Data from the aviation surveillance system ADS-B confirmed the aircraft stayed at 37,000 feet and flew clean over the runway where it was meant to land.
The plane stayed on the runway for about two and a half hours before departing for the next flight.
Posting about the event on Twitter, aviation analyst Alex Macheras described the event as "deeply concerning".
"Ethiopian Airlines Boeing 737 #ET343 was still at cruising altitude of 37,000ft by the time it reached destination Addis Ababa," he wrote.
"Why hadn't it started to descend for landing? Both pilots were asleep."
Macheras blamed pilot fatigue for the incident.
"Pilot fatigue is nothing new, and continues to pose one of the most significant threats to air safety - internationally," he added.