While Denmark's Björn was making steady progress with 13 consecutive pars to remain 11 under par, big-hitting Quiros was defying testing conditions on the Doha course.
The Spaniard, first and second on his last two appearances here, got the bit between his teeth with a birdie at the fifth.
But it was after the turn where he really turned on the pyrotechnics - reaching the par five tenth in two to set-up a birdie, picking up another shot at the next and hitting his approach at the 14th within two feet to make it three gains in five holes.
That left him ten under, one behind Björn and two ahead of Austrian Markus Brier, who after a 25 foot birdie at the first had seen his challenge unravel with four dropped shots before the turn.
Quiros missed a four footer for par on the 15th – his first dropped shot since the second round – and when Björn hit his six iron approach at the same hole within five feet the advantage was back with the former Ryder Cup star.
Björn holed his birdie effort, and a two shot swing left the ten-time European Tour winner three clear with only three to play.
Former Austria Golf Open winner Rafael Cabrera Bello joined Brier on eight under with an eight foot birdie at the 16th.