They spotted the gadget, which had multiple arms and up to eight rotors, on the approach to Heathrow Airport in west London, the UK Airprox Board (UKAB) said.
They both exclaimed: "Was that a drone? At 10,000 feet!"
There was "no time to react" as they watched the flying object for around five seconds as it passed down the right side of the plane, according to the report.
Police officers at the airport were alerted to the near-miss, but the drone operator could not be traced.
The pilots estimated the separation between the aircraft and the drone to be 100 feet (30 metres) vertically and 200 metres (656 feet) horizontally.
The UKAB concluded that "safety was not assured" during the incident on November 11 last year, and determined the risk to be the second most serious category.
It was one of four near-misses between aircraft and drones in the latest monthly UKAB report, bringing the total over the past 12 months to 59.
Civil Aviation Authority rules state that drones must not be flown above 400 feet or near airports or airfields.
In November it launched a website to publish its revised code of conduct for drones, called the dronecode.
Ministers are considering mandatory registration for new drones to crack down on reckless users.
The proposal is part of a Department for Transport consultation on improving drone safety. It is hoped the scheme could help authorities identify the owners of drones being flown illegally.