Hibernian will tackle First Division side Falkirk, the 2009 finalists, in the other tie as the SPL outfit continue their quest to end a 111-year wait to get their hands on the prestigious trophy.
Dundee United, 2-1 winners over Dundee on Sunday, have won the trophy twice, the last occasion being when they beat Ross County at Hampden Park in 2010.
Celtic captain Scott Brown said: "Dundee United in the cup will be a hard game as they have started to hit some form but I have faith in our lads."
The ties will be played on the weekend of April 13 and 14.
The draw was made by former Scotland international and Celtic assistant manager John Robertson on Monday, aided by SFA president Campbell Ogilvie and Lucy Carter, head of brand at William Hill.