Miller's goal handed Gary Megson his first victory since taking over as manager.
Wednesday frustrated a Carlisle side lacking in creative ideas and it was a relatively easy afternoon for keeper Nicky Weaver, who had just a couple of saves of any note to make.
Carlisle rang the changes in the second half but their new strikeforce of Rory Loy and Paddy Madden had no more joy than their predecessors and Wednesday looked the more likely to score again when Gary Teale scooped a good chance over the bar.
Apart from an early scare for Owls, when Mark Beevers had to step in to prevent Craig Curran getting a clear shot at goal, the lion's share of first-half possession went the visitors way.
Gary Madine, sold by Carlisle to Wednesday in the January window, had a hand in the visitors' 20th-minute goal when he flicked on Tommy Spurr's right-wing corner. And although Matty Robson was back on the goalline, his attempted clearance fell to Miller six yards out and he crashed an unstoppable shot into the roof of the net.
Weaver was largely a spectator until the 35th minute when he was called upon to produce a spectacular save from Ben Marshall's curling shot.