Honours finished even following a low-key contest that did little to warm the small crowd that braved the freezing conditions.
Cork who led at the break appeared set for the win as the contest came to a conclusion only for Drogheda to deservedly nick a point from the penalty spot with three minutes remaining.
Against the run of play City broke the deadlock on 33 minutes when Shane Duggan saw his initial effort charged down, however, he met the ball for a second time driving it past Gabriel Sava to the bottom corner of the net.
With 13 minutes remaining Cork almost doubled their lead as Gearoid Morrissey picked up a poor clearance from Sava and returned the ball towards the target with a powerful drive that shaved the post.
The home side eventually earned a deserved point when they won a penalty on 87 minutes after John Dunleavy handled Peter Hynes’s effort and Declan O’Brien converted the resulting spot kick.