The Bhoys suffered a major injury blow just before the game, with Dutch striker Jan Vennegor of Hesselink ruled out.
The game started off in scrappy fashion, with neither side able to forge an opening until the 13th minute, when Shunsuke Nakamura found space at the far post, but Michael Gravgaard made a superb block on the right-foot shot.
Eight minutes later the impressive Aiden McGeady fired in a fine cross from the left, but Miller couldn't direct his header on target.
Maciej Zurawski then had an effort ruled out for offside, before the Bhoys got the chance to break the deadlock ten minutes before the interval.
Nakamura went down under the challenge of Gravgaard and Miller rifled in the spot-kick despite former Rangers keeper Jesper Christiansen getting a hand to the effort.
It was almost 2-0 right on the stroke of half-time, as a superb move culminated in Lee Naylor curling in a excellent cross from the left to the near post and Miller's header was kept out brilliantly by Christiansen.
After the restart, Michael Silberbauer headed over from Lars Jacobsen's cross in from the right, but the home side remained in control without forging any more openings.
Just past the hour Bo Norregaard forced a first save of the game from Artur Boruc with a left-foot drive from just outside the area.
Gordon Strachan's side tried to play keep-ball as the clock ticked on, with the busy Miller always looking to get a second goal for the SPL champions.
In the final minute of the game a far-post free-kick was headed back to Atiba Hutchinson, but his looping header dropped straight into the arms of Boruc and the final whistle sounded seconds later.