Neil Lennon's side survived a late Dunfermline Athletic equaliser to advance to the last eight with a 2-1 victory after extra-time.
They'll face former Parkhead centre-back Gary Caldwell's Partick Thistle side, who defeated Ross County on Saturday.
Rangers travel to Livingston in the next round after defeating East Fife 3-0 on Sunday afternoon. The West Lothian hosts were able to avoid a similar League One banana skin, defeating Forfar Athletic 2-1.
Hearts and Aberdeen will battle it out at Tynecastle in what is arguably the tie of the round. Hearts knocked out Motherwell on Friday night while Aberdeen were able to move beyond Dundee, thanks in large part to an injury-time equaliser by Andrew Considine.
Another highly-anticipated encounter comes at Rugby Park where Angelo Alessio's men will face Hibs. Both teams required extra-time to advance on Saturday as Killie defeated Hamilton 1-0 and Hibs survived a ding-dong battle with Greenock Morton, eventually prevailing 5-3.
Celtic v Partick Thistle
Hearts v Aberdeen or Dundee
Livingston v Rangers
Kilmarnock v Hibs
Ties are due to be played the midweek of September 24-26.