Neither side managed to find the net in the allocated 90 minutes but Sligo were reduced to 10 men when Daniel Ledwith was red card on 77 minutes.
Derry made the numerical advantage count scoring two goals in extra time through Seamus Sharkey and Mark Timlin.
Elsewhere, Ulster Senior League Champions Cockhill Celtic lost 4-2 to Galway Utd at Eamonn Deacy Park.
The Inishowen side did score the games first goal on 4 minutes, Jimmy Bradley netting a fantastic effort from 40 yards out.
With the game tied at half time 1-1, the League of Ireland side took control in the second half scoring 3 goals to progress to the next stage.
Cockhill did score a late consolation which was another cracking goal from Mark Moran.