Alex Burke scored the only goal of the game in the 21st minute after he was put clean through by Steve McGarry. The former Queen of the South striker kept his composure, going around Craig Nelson to slot the ball home.
County made a great start and Marciej Nuckowski missed a great chance to put the home side ahead in the third minute.
In Brechin's first attack Steven Hampshire tried his luck from 15 yards, but his goalbound shot was deflected over by Martin Canning.
But it was mostly County and Nelson made a brilliant one-handed save when Sean Webb looked certain to head in a corner.
A goal looked on the cards and it duly arrived in the 21st minute when an excellent through ball from McGarry picked out Burke, who raced through on goal and rounded the keeper to score.
Brechin were finding it hard to make an impression, and they had only a long shot from former Kilmarnock player Ally Mitchell to show for their efforts.
County kept up the pressure on and Canning headed over from another corner.
After a slow start to the second half there were chances at either end. County's John Rankin had a low drive blocked by Brechin full-back Craig McEwan, and at the other end Paul Ritchie was denied by Canning.
County substiute Sean Higgins failed to control a bouncing ball 10 yards out as the home side went for a killer second.
Brechin hit back and Country were living dangerously. Scott Walker headed just over from a corner. But County held out, and Winter could have doubled the lead in injury time but his curling shot was well saved by Nelson.