Dumbarton struggled to contain Rangers, but had a goalscoring opportunity of their own in the 14th minute as Garry Fleming gained possession in the danger zone and dragged a shot wide of Steve Simonsen's post.
Rangers started the second period brightly and should have extended their lead in the 54th minute when Kris Boyd picked out Miller with a through ball, only for Rogers to come to Dumbarton's rescue and deny the striker at point-blank range.
It took the visitors until the hour mark to extend their lead, with Wallace going on a surging run before exchanging passes with Miller and shooting across the keeper to put his side in control.
McCoist's men made the game safe in the 72nd minute courtesy of a Boyd strike, with the veteran nodding home a Smith cross from the left to put the away side out of sight.
McCulloch almost made amends for his penalty miss by burying a free kick late on, but Rogers produced a good stop to keep the deficit at 3-0 come the final whistle.