Brechin had an early chance after seven minutes when Iain Russell shot just wide of the far post.
Cowdenbeath went close after 14 minutes when a drive from Kevin McBride took a huge deflection off a defender and came back off the post.
After 20 minutes Craig Nelson saved with his feet from Pat Clarke and Liam Buchanan should have netted the rebound but put the ball over the bar.
Soon after a solo run from Kevin Byers ended with a shot which was saved by David Hay.
The deadlock was broken after 38 minutes when Hill rose to head home a McBride corner.
Cowdenbeath looked for more in the second half when Nelson fumbled but Hill drove wide.
Gary Fusco should have scored just before the hour mark but blazed over then Clarke missed the target with a close range header.
Clarke had another chance when he sent a dipping drive just over the bar.
After 69 minutes Byers had a 25-yard drive well saved by keeper Hay.
A minute later Darren Smith crossed and a first-time effort from Steven Hampshire was saved by Hay.
Brechin's final chance came when Byres drove over from 20 yards.