Rangers will play either Swedish side Malmo or Faroe Islands champions Torshavn in the third qualifying round of the Champions League.
Malmo are currently in pole position to take on Ally McCoist's side after a 2-0 victory in the first leg, with a trip to Torshavn to come on Tuesday.
It was certainly not the kindest draw Rangers could have had for the two-legged tie, which will begin with a home match on July 26/27, with the away leg taking place the following week.
Clarke finished in style with a birdie on the 18th to complete a second successive 68, sharing the clubhouse lead on four under with Glover (70).
Bjorn had fallen back to three under with bogeys at the second and third, leaving playing partner Simon Dyson to charge to the head of the field with an opening hat-trick of birdies.
Lee Westwood remained four over par with four to play, the world number two in severe danger of missing the cut after five top-three finishes in his previous seven major championships.