The home side went close after 13 minutes when Gerry Britton took a return pass from Steven Hampshire and shot inches wide of the post.
Dundee opened the scoring after 19 minutes when Kevin McDonald drove the ball home after Garry Brady had a shot blocked.
After 22 minutes Britton laid the ball off to Hampshire, and his drive forced a great save from Scott Murray.
Dundee went close a minute later when Bryan Deasley knocked the ball back to Scott Robertson, and he shot just wide. In the 32nd minute, Daniel Grainger drove wide from 25 yards.
Brechin pressed as half-time approached and a fine shot from Charlie King was saved by Murray. Then Hampshire fired a drive just over from the edge of the box.
A minute into the second half Simon Lynch shot wide after a great run from Kevin McDonald.
Dundee made in 2-0 after 53 minutes when Paul Dixon scored with a superb swerving shot from 25 yards which came down off the underside of the cross bar and struck Craig Nelson before going in.
Dundee stretched their lead to three in the 62nd minute when Lynch was brought down by Nelson and got up to score from the penalty spot.
Six minutes later Dixon put a free-kick just wide of the post then, as Dundee continued to dominate, Deasley sent two headers wide of the target.
Brechin were denied a consolation goal after 85 minutes when a lob from Darren Smith was fumbled by Murray and dropped just wide of the Dundee goal.