Veteran Bjorn, who was in joint-third position overnight, finished Saturday's round with a sensational hat-trick of birdies starting on the 16th hole, to set himself up for a tilt at an 11th European Tour title.
Bjorn, who won the Portuguese Open last year to claim his first title in four years, took advantage of two bogeys by Brier to post a total of 205.
Swede Robert Karlsson, the defending champion, was in third spot, three shots behind on 208, while South African Thomas Aiken occupied fourth position on 209.
Former winner Alvaro Quioros of Spain, and England's Richard Finch were a further shot back on 210.
Bjorn spent four winless years on the European Tour after his victory at the Irish Open in 2006, but his sensational five-shot win in Portugal last year not only revived his career, but also helped him become one of Colin Montgomeries's vice-captains for the Ryder Cup campaign.
South African Darren Fichardt, who was in second spot after the second round, slipped down to a tie for 11th position, after four bogeys ruined his third round.
Meanwhile, Martin Kaymer, who was afforded a slim chance of becoming the new world number one after Lee Westwood failed to make the half-way cut, shot a four-under-par 68 for an aggregate score of 215.
But his first-round score of 77 meant he barely managed to make the cut, and with Bjorn leading him by 10 shots, he will have to wait a little longer if he is to become only the second German to top the world rankings after veteran Bernhard Langer.
Leading third-round scores from the Qatar Masters
Thomas Bjorn (Denmark) 74 65 66
Markus Brier (Austria) 71 66 69
Robert Karlsson (Sweden) 70 69 69
Thomas Aiken (South Africa) 71 69 69, Alvaro Quiros (Spain) 75 69 66
Richard Finch (Britain) 70 69 71,
Rafael Cabrera (Spain) 77 68 66
Steven O'Hara (Britain) 73 69 69
Louis Oosthuizen (South Africa) 73 69 69
Paul Lawrie (Britain) 74 66 71 212 Gregory Havret (France) 73 73 66
Mark Foster (Britain) 73 71 68
David Howell (Britain) 75 69 68
Darren Fichardt (South Africa) 70 68 74 213 Soren Hansen (Denmark) 74 71 68
Felipe Aguilar (Chile) 77 66 70
David Lynn (Britain) 73 69 71
Victor Dubuisson (France) 74 68 71 214 Ricardo Gonzalez (Argentina) 73 74 67
Keith Horne (South Africa) 78 69 67
Stephen Gallacher (Britain) 72 73 69
Graeme Storm (Britain) 75 70 69
Anthony Wall (Britain) 75 70 69
Kenneth Ferrie (Britain) 75 69 70
Oliver Wilson (Britain) 74 69 71
Robert Rock (Britain) 73 68 73
David Drysdale (Britain) 71 71 72