Brechin were first to threaten after 2 minutes when Iain Russell was in on goal but Paul Mathers dived to save.
On 10 minutes the ball broke to Stuart Callaghan and his chip just cleared the crossbar.
At the other end Bobby Linn made space for himself but shot straight at Craig Nelson.
On 16 minutes Charlie King drove just wide from a Darren Smith pass.
On 22 minutes Peterhead hit on the break and Stuart McKay shot over from a Graeme Sharp cross.
Back came Peterhead and on 36 minutes Linn headed wide from a Sharp cross.
Brechin opened the scoring on 42 minutes when the referee played a great advantage following a foul by Craig Tully and Charlie King ran in to score. Before play resumed Tully was yellow carded.
Peterhead began the second half well and Bryan Gilfillan should have done better than drive over then on 54 minutes Keith Gibson had an effort saved on the near post.
Brechin had a chance when David White stabbed the ball towards goal and Mark Perry cleared off the line.
Goalscorer King shot into the side-netting and at the other end a net-bound effort from Gilfillan was blocked by the home defence.
Craig Nelson was the home hero 10 minutes from time when he made a superb save from substitute Martin Bavidge.
Peterhead pressed right to the end and on 85 minutes Martin Wood should have done better than head over from a Gilfillan cross.