The Dark Blues have now gone nine league games without a win since late January and made little impression against a team which they had already beaten three times this year.
All of Dundee's recent frustrations reached a new level as City took a shock lead despite being bombarded in the opening 10 minutes.
From a long throw by Daniel Grainger, much travelled striker Alex Burns punished Dundee's wretched defending by directing a looping header into the far corner of the net.
It was the first goal conceded by teenage keeper Scott Murray in seven games and ended his hopes of beating a 20-year club record of clean sheets.
Two minutes later a bungled pass back from Robert Mann left Murray in trouble again but on this occasion Burns slammed the ball into the side-netting.
Darren Smith also went close for Brechin as Dundee dominated possession but lacked a real cutting edge up front.
Shortly before the break Stuart McCluskey's header crashed back off the bar and the follow up shot from Kevin McDonald was brilliantly blocked by veteran Brechin keeper Craig Nelson.
Eighteen goal striker Simon Lynch finally joined the Dundee attack in the second period but the clearer chances continued to fall to the visitors with Steven Hampshire, David White and Burns all failing to score the second goal which they felt they needed but in fact didn't require.