Celtic, the holders, have travelled to Peterhead and Inverness Caledonian Thistle in the previous two rounds and now face a trip to Tannadice.
Peter Houston's side lie seventh in the SPL table this season but have finished fourth and third the last two years.
The Arabs beat Celtic's Old Firm rivals Rangers 2-0 on Sunday to reach the last eight and will provide a stern test for Neil Lennon's men.
Monday afternoon's draw also threw up a potential clash between Motherwell, currently third in the SPL, and Aberdeen should the sixth-placed Dons come through their replay with Irn-Bru Division One side Queen of the South.
St Mirren also face an all-SPL tie if they come through their replay against Ross County, with the winners of that tie drawn away to Hearts or St Johnstone.
Hibernian travel to Ayr United or Falkirk, who will contest their fifth-round clash on February 15 after Saturday's clash was postponed when the Somerset Park pitch failed a morning inspection.
The three fifth-round replays will be played a day earlier on February 14, while the quarter-finals will take place on the weekend of March 10 and 11.
Quarter-final draw:
Hearts/St Johnstone v St Mirren/Ross County
Ayr United/Falkirk v Hibernian
Dundee United v Celtic
Motherwell v Aberdeen/Queen of the South
Ties to be be played weekend of March 10-11.
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