The Bairns were frustrated for long spells by ten-man City but they ended the day nearer to the promotion play-off places than the relegation ones.
Falkirk made a lively start with Rory Loy and Jakubiak having efforts on goal, setting a pattern which was to endure.
Jakubiak, who has impressed since joining on loan from Watford, might have opened the scoring on 22 minutes but his 18-yard shot rose just too high.
A Reghan Tumilty effort saw a similar outcome as Brechin defended resolutely, largely restricting their rivals to shots from distance.
The home side were rarely seen as an attacking force and their hopes of a clinging on for a point received a setback midway through the second period when Aaron Lynas was shown the red card for a bad foul on Tom Taiwo.
Falkirk substitute Joe McKee twice went close as the visitors attempted to make the extra man count.
Just as it looked the game would end all square, Jakubiak popped-up in a crowded box to stab home.