County were 3-0 up through goals from Kevin McKinlay, John Rankin and Alex Burke before Andy Thomson grabbed a consolation for Queens.
County looked sharp at the start, and Rankin tried a low shot that Ritchie Barnard gathered. Then a sweeping cross from David Cowie was snatched out of the air by the Queens keeper before Burke could pounce.
At the other end, Paul Burns took a fresh air swipe at the ball outside the County box before it was cleared.
When Lionel Djebi-Zadi's chip into the box was only cleared to Cowie, his shot was well saved by Barnard. Then Burke caught the keeper off his line, but his cross to the far post was missed by Sean Higgins.
But the goal did come in the 26th minute when Burke played the ball in through a ruck of legs and McKinlay tapped in at the back post.
Jim Lauchlan rose high in the box to get on the end of a Cowie free-kick in the 34th minute, but his header finished well over.
Willie Gibson and Steve Bowey both came on for Queens at the start of the second half, replacing Graham Weir and Brian Reid.
But it was County who made the best possible start when Burke found Higgins on the edge of the area and his lay-off was fired into the net by Rankin.
McKinlay should have put the issue beyond doubt in the 66th minute when he was clean through on goal, but he sent his shot into the side netting.
County stretched their lead six minutes later when a Cowie corner was cleared as far as Martin Canning, whose shot was re-directed into the net by Burke's head.
But Queens were not finished and they got one back in the 74th minute when a Gibson cross was headed into the corner of the net by Thomson. The visitors almost got another when Gibson tested Stuart Garden with a vicious shot that the keeper did well to keep out.
FootyMAD verdict - A jaded performance from a Queens team obviously suffering from that gruelling extra time cup tie at Hamilton in midweek. The Queens defence never really got to grips with the lively County attack superbly let by ex-Queens favourite Burke. Manager McCall changed the formation and had the home side on the back foot in the latter stages of the match but by that time it was too late. Queens Thomson looked very sharp and fit.
FootyMAD Queens man of the match - Andy Thomson