The Northern Irishman picked up two shots on the back nine to go to four under but that still left him five adrift of the Dane.
But it was again a mixed bag for McIlroy whose struggles of the first two days continued as he picked up four shots in all but also dropped two with a double-bogey six at the 444-yard sixth.
Having picked up a shot at the par three second he then eagled the 586-yard fourth before throwing away what he had gained two holes later.
A double-bogey six on the sixth was followed by a birdie on the seventh as McIlroy went out in 34 as he looked to improve on shooting 70 on each of the first two days.
Bjorn made a steady start by parring his first three holes to retain the lead on nine under, the Dane one shot ahead of England's Chris Wood who picked up one stroke with a birdie on the second.
South African George Coetzee was alone in third on seven under and Italian Francesco Molinari shared fourth a shot further back with Thailand's Chapchai Nirat.