The Binos appeared to be coasting to victory when they surged into a two-goal lead soon after half-time but needed a late penalty save from Myles Hogarth to earn them all three points.
With just five minutes left, Stewart Devine was adjudged to have impeded fellow substitute Chris Geddes and referee Mike Ritchie awarded a spot-kick. Stuart Callaghan took it but his effort was too straight and Hogarth saved with his legs, then recovered to flip Darren Smith's follow-up header over the bar.
Albion were the better team in the first half and took a deserved lead in 19 minutes when Colin Cramb picked out Steven Bell, who took a touch then thumped a low 20-yard shot just inside the post.
Nathan Taggart was denied a second goal when Craig Nelson made a smart save three minutes later but Albion did double their lead four minutes after half-time with a rare goal from a corner.
Chris Aitken's cross carried as far as Cramb who smashed a superb shot in off the bar for his seventh goal of the season.
Brechin looked down and out but got themselves back into the game when a mistake from Sean Roycroft allowed Steven Hampshire to set up Smith, who swept the ball home after 53 minutes.
Paddy Connolly then had a potential equaliser ruled out for offside before Hogarth broke City hearts with his penalty save as Albion enhanced their position in second place.