The draw for the last four was made live on BBC One Scotland's Sportscene with United coming out the hat first, followed by the Dons.
The ties at neutral venues will be played on 31 January and 1 February.
Rangers were the first side into the semis by beating St Johnstone on Tuesday, with Aberdeen, Celtic and United winning the following night.
The Dons beat Hamilton 1-0, Celtic thrashed Partick Thistle 6-0 and the Tangerines won 7-6 on penalties after a 3-3 draw with Hibernian.
Scotland's national stadium, Hampden Park, is likely to host both semi-finals.
Celtic last met Rangers in a 2012 league match, when the former triumphed 3-0. The Ibrox side won their last League Cup meeting - the 2011 final.
Reacting to the draw, Celtic manager Ronny Deila told BBC Scotland: "I haven't been at an Old Firm game before but I am looking forward to experiencing it.
"Of course we are favourites. We are favourites in every game here in Scotland.
"Everybody knows it is unbelievably important in Glasgow. Everyone is looking forward to a fantastic atmosphere. It is two clubs that really want to beat each other.
"You can hear from the dressing room that people are looking forward to it."
Asked if he had wanted to be drawn against the Ibrox side, the Norwegian replied: "Yes, it is no problem for me. There were three good teams there and I think we didn't get the best one."
And Rangers manager Ally McCoist told his club website: "I'm delighted with the draw and judging by the reaction from the boys in the dressing room they're happier than me.
"It's a great draw for us. Everyone has anticipated it and I don't think there's any doubt Celtic will start as favourites. Of course they will.
"It's a smashing game for us though. We feel the boys have come a long way and we're now in a position where we feel we can compete with teams on our day.
"We're not going to get carried away but I think it says a lot about the last couple of years that we're now looking forward to that type of game.
"It can't be a distraction though and it absolutely won't be. It's a League Cup semi-final with Celtic but nothing more.
"Whether anybody likes it or not, it isn't as high on the agenda as getting out of the division (the Scottish Championship) is for us."