The last-four ties will be played on the weekend of 10 and 11 April.
Celtic booked their place with a 3-0 win over Kilmarnock, while Raith Rovers saw off fellow First Division outfit Dundee at Dens Park.
Ross County fought back to draw 2-2 with Hibs at Easter Road at the weekend. The sides will meet in a replay on Tuesday 23 March, with United and Rangers in action the following evening.
Active Nation Scottish Cup semi-final draw
Celtic v Hibernian or Ross County
Rangers or Dundee United v Raith Rovers
Source: BBC Sport
Source: BBC Sport