Cork made it four 1-0 wins on the trot thanks to goalace Pat Morley's late strike.
The goal came after 78 minutes of a tight match played in diffcult, wet conditions.
Alan Carey got forward to whip over a cross to the far post where substitute Morley arrived to head home his 179th league goal to move him up to second place in the all-time scorers chart.
The win moves City, unbeaten in five matches under Liam Murphy, up to fifth place in the table and well in contention for a European place.