Brechin slipped five points off the League 1 pace after a day to forget at the national stadium.
A first-half penalty from on-loan Hearts striker Dario Zanatta and a second-half header from Ryan McGeever left player/manager Darren Dods ruing his side’s “worst performance of the season”.
City remain in third place but Dods said: “They looked hungrier than us, and that is disappointing. We were sluggish and never created enough.
“The display was not like us.”
He added: “When you come to Hampden you are playing on the best pitch you are likely to play on all season. That should have perked us up, but we have put in our worst display of the season.”
The game turned six minutes from the break when the home side won a penalty after keeper Smith brought down Zanatta.
Ross Millen took the spot kick and drilled past the shot-stopper, with both players then booked after an angry exchange between them.
City have the chance to make amends when they entertain Albion Rovers tomorrow, and Dods added: “It is a quick turnaround and we need to be much sharper in that game.”