Kevin McBride hit the post for the visitors after 15 minutes before Liam Buchanan spurned a great chance to give them an early advantage.
The visitors broke the deadlock after 38 minutes when defender Douglas Hill rose to head home McBride's corner.
Brechin pressed for an equaliser to force extra time but could not breach Cowdenbeath's defence.
The home side missed an early chance when Iain Russell shot just wide of the post.
Cowdenbeath went close in the 14th minute when a drive from Kevin McBride took a huge deflection off a defender and came back off a post.
Then home goalkeeper Craig Nelson saved from Pat Clarke and Buchanan put the rebound over the crossbar in the 20th minute.
Cowdenbeath continued to be the better side in the second half and Gary Fusco missed a great chance before Clarke headed wide from close range.
Brechin: Nelson, Archibald, Ward, White, Callaghan, Byers, Steven Ferguson (Stuart Ferguson 85), King (Connolly 54), Smith, Hampshire, Russell (Geddes 72). Subs Not Used: Flynn, Hillcoat.
Cowdenbeath: Hay, Baxter, Lennon, Hill, McBride, Patrick Scullion (Fyfe 76), Hannah, Fotheringham (Dalziel 72), Fusco, Buchanan, Clarke. Subs Not Used: Weir, Marshall, Allison.
Goals: Hill 38.
Att: 594
Ref: B Winter