A late Dennehy penalty ensured that the Leesiders kept their unbeaten record intact in a fiercely contested game at Turner's Cross
An intense game started off slowly as play was stopped early into the game when Garry Buckley collided with Sligo Rovers goalkeeper Gary Rogers while challenging for a cross.
The stoppage in play took the wind out of Cork's sails as Sligo upped their tempo with wave after wave of attack. Indeed, the visitors could have gone ahead after 15 minutes when Greene latched onto a clever flick from Joseph Ndo in the box, but his shot was blocked by John Dunleavy.
Cork answered with some swift counter-attacking football and should have taken the lead minutes later when Mark O'Sullivan eluded the attention of Jeff Henderson, but the former Avondale striker's shot was blasted over the bar.
A frenetic second half yielded plenty of chances for both teams with Cork threatening to snatch the lead immediately after the re-start through Dan Murray, but Evan McMillan was on hand to block the header.
John Caulfield reshuffled his pack in search of a winning goal sending Rob Lehane into the fray to partner O'Sullivan in attack. The young striker injected extra energy into the Cork front line and almost put his team ahead with a powerful effort, but Rogers was equal to it.
Sligo profited from Cork's cavalier approach and took the lead in the dying moments when Kieran Djilali prodded home after some frantic play following Joseph Ndo's corner.
However, Sligo's lead was short-lived as Kalen Spillane knocked Mark O'Sullivan over in the box and Billy Dennehy confidently dispatched past Rogers to clinch a draw.
The Bit O'Red return to action on Monday against St Patrick's Athletic in the Setanta Sports Cup, while Cork City's next game sees them host Drogheda United on Friday. Dundalk's win over UCD has seen them overtake the Leesiders at the top of the table, but Cork's unbeaten start means they are just two points off the summit.