The pair stayed ahead of the chasing pack as they both carded third-round 68s at the Pretoria Country Club to finish on 13 under par.
Bjork is playing in just his ninth European Tour event but showed the composure of a veteran as he made three birdies on the front nine and recovered from two bogeys to birdie the last hole.
Jamieson also carded five birdies and two bogeys, coming close to an eagle on the last that would have given him the overnight lead.
They were a stroke ahead of Dean Burmester of South Africa and Jorge Campillo of Spain with Mikko Korhonen of Finland a further shot back.
Englishman Oli Fisher had the best score of the day, hitting nine birdies and nine pars for a first ever 62 in 865 European Tour rounds. He was among five players on 10-under.
Masters champion Trevor Immelman, who made his first cut in 12 tournaments, hit a seven iron from 201 yards at the par three fifth for a hole-in-one. He was on three under par going into the final round.
(Reporting by Mark Gleeson in Cape Town; Editing by Ed Osmond)
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