United pulled off a shock by defeating Rangers 2-0 at Ibrox yesterday to book their place in the quarter-finals.
Hearts or SPL rivals St Johnstone will host the winners of the replay involving St Mirren and Ross County, while Hibernian face an away match against First Division opposition in either Ayr United or Falkirk, whose game was postponed.
Motherwell have been drawn against either Aberdeen or Queen of the South, who fought out a 1-1 draw at the weekend.
The ties will be played on Saturday 10th or Sunday 11th March.
The draw in full:
Heart of Midlothian or St Johnstone v St Mirren or Ross County
Ayr United or Falkirk v Hibernian
Dundee United v Celtic
Motherwell v Aberdeen or Queen of the South