The Dubai Grand Prix of 1981 was never an official Grand Prix as such....
from Grandprix.com:
There has long been talk of races in the Middle East but the only major event was in Dubai in December 1981 when British businessman - Martin Hone, who later organized the Birmingham Superprix - organized a Grand Prixon a makeshift 1.6-mile track around the Hyatt Regency Hotel. There were five events and considerable appearance money meant that the turnout of VIPs was impressive. There was talk at the time of the track being extended but the Dubai GP never happened. The Arab world became more interested in rallying with the highly-successful Middle Eastern Rally Championship.
and this news strongly indicates that there was a 'race' for the first official Grand Prix, and Bahrain has won it:
DUBAI is planning to build a Formula 1 circuit in the course of the next 18 months as part of a new development which will be called the Dubai Autodrome and Business Park. The facility will feature kart tracks, a driving school and tracks for motorcycle racing in addition to a three-mile road course which is intended to be up to the latest F1-spec. The scheme is the brainchild of Union Properties. The intention is for the facility to become the center of the motorsport industry in the Middle East. There is expected to be a street race in Dubai at the end of this year to help add to the momentum.
Couldn't help noticing......core sponsor of the 81' Dubai GP was Gulf Air....my how the times change!!!