After a fairly even first half, City broke the deadlock in the 62nd minute with at terrific goal from ex-Alloa player Stuart Callaghan who volleyed home from 25 yards giving keeper Allan Creer no chance.
The visitors tried hard to rescue a point but another wonder goal from another ex-Alloa player wrapped up the points for City in the final minute.
Creer came off his line to clear the ball away but kicked it out to Brechin substitute Richard Walker 35 yards from goal and the midfielder took one touch before coolly lobbing the big keeper.
Brechin boss Michael O'Neill said: "I am delighted with the win. The players are playing some good stuff at the moment and enjoying themselves.
"Our recent results have been a wee bit patchy and we now have to concentrate on getting some consistency into our results."
Alloa boss Allan Maitland said: "We had to work hard to match Brechin. I always felt that a mistake or a wonder goal would win the match and so it proved."