David Ross and Barry Wilson both had chances to equalise for the home side but failed to convert.
Charlie King doubled Brechin's lead from a Rory McAllister pass before McAllister netted himself late on to seal the points for the visitors.
Peterhead: Jarvie, Donald, Strachan, Crawford, Stuart Smith, Barry Wilson, McVitie, Clark (Emslie 10), Sharp, Ross (Bruce 80), Bavidge. Subs Not Used: Keiran Wilson, Bowden, Young.
Booked: McVitie.
Brechin: Nelson, Richard Walker, Smith, McLean, Dyer, Byers, Janczyk, Fusco, Docherty (Canning 86), McAllister, King (Cowan 89). Subs Not Used: Scott.
Booked: Smith, Richard Walker.
Goals: Byers 31, King 75, McAllister 84.
Att: 553
Ref: A Law
Source: BBC Sport
Source: BBC Sport