The flight at a test range in the US state of Oregon lasted just under one minute, with the aircraft lifting up five metres from the ground and landing without control from a pilot.
Airbus says the jet is the world's longest-range single-aisle aircraft, capable of flying up to 4,000 nautical miles when configured with 206 seats including lie-flat beds.
Qatar Airways group chief executive Akbar al-Baker welcomed the arrival of the Airbus A350-1000 test aircraft to Doha, as part of a special demonstration tour by Airbus.
Qatar Airways will take delivery of the first Airbus A350-1000 aircraft between February 15 and 20, group chief executive Akbar al-Baker has said.
Qatar Airways tried to buy up to 10% of listed American Airlines last year, but changed its mind in August after opposition from the US carrier's management.
Airbus's decision in 2007 to pursue the A380, capable of packing in 853 seats, was diametrically opposed to Boeing's bet on the Dreamliner, marketed as a more efficient plane.
Frenetic selling in the closing weeks of 2017 saw Airbus overhaul Boeing's recent lead in the global jet market to win their annual order contest for the fifth year running, while remaining behind its US rival in the number of planes built.
The potential A320 order, which had not been previously announced, would be worth more than $18bn at list prices.