With the sale of Nikica Jelavic to Everton in midweek all the focus before the game was on the Rangers forward line but it was United's attackers who dominated with an impressive first-half display.
Jonny Russell had an effort ruled out as early as the second minute when Jon Daly fouled keeper Allan McGregor in the build-up before Gavin Gunning headed United into a 15th minute lead.
Sone Aluko and David Healy came close to an equaliser before Russell doubled United's advantage in the 35th minute when he finished a flowing move.
Rangers manager Ally McCoist used all three substitutes in an attempt to find a way back into the game, with both sides picking up five bookings apiece as tempers flared, but keeper Dusan Pernis didn't have a save to make in a quiet second half.
McCoist felt it had been a disappointing defensive display.
"You have to give credit where it is due and I thought United, especially in the first half, played well," McCoist, whose side now only have the SPL title to play for, said.
"But in terms of our defending for both goals it was really poor and I think it was far too easy. That is part of our game that we do take pride in but it was poor today."
His United counterpart Peter Houston hailed a fantastic day for his club.
"What I was worried about at the start was a wounded Rangers coming out and taking the game to us but I don't think we gave them that opportunity," Houston said.
"Right from the off we took the game to Rangers and deservedly scored a couple of good goals. In the second half we stopped Rangers from creating the chances we know they can do and got in their faces and stopped them playing."
It is the second time in three years United have knocked the Glasgow giants out of the competition and Houston's side will join treble-chasing Celtic, who dispatched Inverness Caledonian Thistle on Saturday, in Monday's quarter-final draw.