The league leaders started brightly and Craig Nelson held a Chris Millar drive in the first minute.
Brechin settled and Stefan Gonet was relieved when John Ward's looping header from Stuart Callaghan's free-kick landed on top of his net.
Nelson did superbly to palm away a Stewart Greacen header from a Jamie Stevenson free-kick, but the big centre-back almost gifted City a goal, allowing Darren Smith through but Gonet made a spectacular save to fist the ball away for a corner.
The game's only goal came seven minutes from the break and was beautifully constructed.
Stevenson released Peter Weatherson on the right flank and when the wing-back's low cross flew across the goalmouth, Derek Lilley deftly played it into the path of James McAlistair who beat Nelson with a low shot from ten yards.
Paul McGowan had a low shot saved after McAlistair's cross was knocked back by Weatherson and, from another cross, Millar's rocket was blocked bravely by Stuart Ferguson.
Ton boss Jim McInally is always ill-at-ease at 1-0 and Charlie King's late swerving shot which beat Gonet but swerved just wide vindicated his anxiety.
An equaliser would have been an injustice and they almost clinched a second when Kevin Finlayson's low shot was tipped away by the industrious Nelson.